[{"data":1,"prerenderedAt":3734},["ShallowReactive",2],{"subject-categories-math":3,"formula-de\u002Fformulas\u002Fmath\u002Ffunctions\u002Fverdopplungszeit":546,"formula-related-de\u002Fformulas\u002Fmath\u002Ffunctions\u002Fverdopplungszeit":3721},[4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,173,174,175,176,177,178,179,180,181,182,183,184,185,186,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545],{"category":5},"algebra",{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":65},"arithmetic",{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":101},"calculus",{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":172},"differential-equations",{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":187},"discrete-math",{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":208},"functions",{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":247},"geometry",{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":312},"linear-algebra",{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":368},"multivariable-calculus",{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":391},"numerics",{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":412},"sequences-series",{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":441},"stochastics",{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":518},"trigonometry",{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"id":547,"title":548,"aliases":549,"body":555,"category":208,"checks":3591,"conditions":1477,"conditionsHtml":3620,"deprecated":3621,"description":3585,"extension":3622,"figure":3623,"figureCaption":3624,"figureHtml":3689,"formulaId":3690,"latex":3691,"latexHtml":3692,"level":3693,"meta":3694,"navigation":3635,"path":3695,"redirectFrom":3696,"related":3697,"seo":3702,"stem":3703,"subject":585,"summary":3704,"tags":3705,"variables":3710,"__hash__":3720},"formulas_de\u002Fde\u002Fformulas\u002Fmath\u002Ffunctions\u002Fverdopplungszeit.md","Verdopplungszeit",[550,551,552,553,554],"Verdopplungszeit Formel","Verdopplungszeit berechnen","wann verdoppelt sich","T_D = ln 2 \u002F ln q","70er-Regel",{"type":556,"value":557,"toc":3584},"minimark",[558,563,716,1077,1150,1452,1510,2089,2093,2205,2444,2569,2690,2987,2991],[559,560,562],"h2",{"id":561},"erklärung","Erklärung",[564,565,566,567,571,572,574,575,683,684,715],"p",{},"Bei ",[568,569,570],"strong",{},"exponentiellem Wachstum"," dauert es immer gleich lange, bis sich der Bestand verdoppelt, egal wie groß er gerade ist. Diese feste Zeit heißt ",[568,573,548],{}," ",[576,577,580,609],"span",{"className":578},[579],"katex",[576,581,584],{"className":582},[583],"katex-mathml",[585,586,588],"math",{"xmlns":587},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[589,590,591,604],"semantics",{},[592,593,594],"mrow",{},[595,596,597,601],"msub",{},[598,599,600],"mi",{},"T",[598,602,603],{},"D",[605,606,608],"annotation",{"encoding":607},"application\u002Fx-tex","T_D",[576,610,614],{"className":611,"ariaHidden":613},[612],"katex-html","true",[576,615,618,623],{"className":616},[617],"base",[576,619],{"className":620,"style":622},[621],"strut","height:0.8333em;vertical-align:-0.15em;",[576,624,627,632],{"className":625},[626],"mord",[576,628,600],{"className":629,"style":631},[626,630],"mathnormal","margin-right:0.1389em;",[576,633,636],{"className":634},[635],"msupsub",[576,637,641,674],{"className":638},[639,640],"vlist-t","vlist-t2",[576,642,645,669],{"className":643},[644],"vlist-r",[576,646,650],{"className":647,"style":649},[648],"vlist","height:0.3283em;",[576,651,653,658],{"style":652},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[576,654],{"className":655,"style":657},[656],"pstrut","height:2.7em;",[576,659,665],{"className":660},[661,662,663,664],"sizing","reset-size6","size3","mtight",[576,666,603],{"className":667,"style":668},[626,630,664],"margin-right:0.0278em;",[576,670,673],{"className":671},[672],"vlist-s","​",[576,675,677],{"className":676},[644],[576,678,681],{"className":679,"style":680},[648],"height:0.15em;",[576,682],{},". Sie hängt nur vom Wachstumsfaktor ",[576,685,687,701],{"className":686},[579],[576,688,690],{"className":689},[583],[585,691,692],{"xmlns":587},[589,693,694,699],{},[592,695,696],{},[598,697,698],{},"q",[605,700,698],{"encoding":607},[576,702,704],{"className":703,"ariaHidden":613},[612],[576,705,707,711],{"className":706},[617],[576,708],{"className":709,"style":710},[621],"height:0.625em;vertical-align:-0.1944em;",[576,712,698],{"className":713,"style":714},[626,630],"margin-right:0.0359em;"," ab, nicht vom Anfangsbestand.",[564,717,718,719,862,863,1006,1007,1038,1039,1076],{},"Die Formel folgt aus der Bedingung ",[576,720,722,753],{"className":721},[579],[576,723,725],{"className":724},[583],[585,726,727],{"xmlns":587},[589,728,729,750],{},[592,730,731,742,746],{},[732,733,734,736],"msup",{},[598,735,698],{},[595,737,738,740],{},[598,739,600],{},[598,741,603],{},[743,744,745],"mo",{},"=",[747,748,749],"mn",{},"2",[605,751,752],{"encoding":607},"q^{T_D} = 2",[576,754,756,852],{"className":755,"ariaHidden":613},[612],[576,757,759,763,840,845,849],{"className":758},[617],[576,760],{"className":761,"style":762},[621],"height:1.0358em;vertical-align:-0.1944em;",[576,764,766,769],{"className":765},[626],[576,767,698],{"className":768,"style":714},[626,630],[576,770,772],{"className":771},[635],[576,773,775],{"className":774},[639],[576,776,778],{"className":777},[644],[576,779,782],{"className":780,"style":781},[648],"height:0.8413em;",[576,783,785,788],{"style":784},"top:-3.063em;margin-right:0.05em;",[576,786],{"className":787,"style":657},[656],[576,789,791],{"className":790},[661,662,663,664],[576,792,794],{"className":793},[626,664],[576,795,797,800],{"className":796},[626,664],[576,798,600],{"className":799,"style":631},[626,630,664],[576,801,803],{"className":802},[635],[576,804,806,831],{"className":805},[639,640],[576,807,809,828],{"className":808},[644],[576,810,813],{"className":811,"style":812},[648],"height:0.3448em;",[576,814,816,820],{"style":815},"top:-2.3567em;margin-left:-0.1389em;margin-right:0.0714em;",[576,817],{"className":818,"style":819},[656],"height:2.5em;",[576,821,825],{"className":822},[661,823,824,664],"reset-size3","size1",[576,826,603],{"className":827,"style":668},[626,630,664],[576,829,673],{"className":830},[672],[576,832,834],{"className":833},[644],[576,835,838],{"className":836,"style":837},[648],"height:0.1433em;",[576,839],{},[576,841],{"className":842,"style":844},[843],"mspace","margin-right:0.2778em;",[576,846,745],{"className":847},[848],"mrel",[576,850],{"className":851,"style":844},[843],[576,853,855,859],{"className":854},[617],[576,856],{"className":857,"style":858},[621],"height:0.6444em;",[576,860,749],{"className":861},[626],". Weil die gesuchte Zeit im Exponenten steht, logarithmiert man beide Seiten: ",[576,864,866,903],{"className":865},[579],[576,867,869],{"className":868},[583],[585,870,871],{"xmlns":587},[589,872,873,900],{},[592,874,875,881,884,887,890,892,894,896,898],{},[595,876,877,879],{},[598,878,600],{},[598,880,603],{},[743,882,883],{},"⋅",[598,885,886],{},"ln",[743,888,889],{},"⁡",[598,891,698],{},[743,893,745],{},[598,895,886],{},[743,897,889],{},[747,899,749],{},[605,901,902],{"encoding":607},"T_D \\cdot \\ln q = \\ln 2",[576,904,906,963,990],{"className":905,"ariaHidden":613},[612],[576,907,909,912,952,956,960],{"className":908},[617],[576,910],{"className":911,"style":622},[621],[576,913,915,918],{"className":914},[626],[576,916,600],{"className":917,"style":631},[626,630],[576,919,921],{"className":920},[635],[576,922,924,944],{"className":923},[639,640],[576,925,927,941],{"className":926},[644],[576,928,930],{"className":929,"style":649},[648],[576,931,932,935],{"style":652},[576,933],{"className":934,"style":657},[656],[576,936,938],{"className":937},[661,662,663,664],[576,939,603],{"className":940,"style":668},[626,630,664],[576,942,673],{"className":943},[672],[576,945,947],{"className":946},[644],[576,948,950],{"className":949,"style":680},[648],[576,951],{},[576,953],{"className":954,"style":955},[843],"margin-right:0.2222em;",[576,957,883],{"className":958},[959],"mbin",[576,961],{"className":962,"style":955},[843],[576,964,966,970,974,978,981,984,987],{"className":965},[617],[576,967],{"className":968,"style":969},[621],"height:0.8889em;vertical-align:-0.1944em;",[576,971,886],{"className":972},[973],"mop",[576,975],{"className":976,"style":977},[843],"margin-right:0.1667em;",[576,979,698],{"className":980,"style":714},[626,630],[576,982],{"className":983,"style":844},[843],[576,985,745],{"className":986},[848],[576,988],{"className":989,"style":844},[843],[576,991,993,997,1000,1003],{"className":992},[617],[576,994],{"className":995,"style":996},[621],"height:0.6944em;",[576,998,886],{"className":999},[973],[576,1001],{"className":1002,"style":977},[843],[576,1004,749],{"className":1005},[626],". Statt ",[576,1008,1010,1026],{"className":1009},[579],[576,1011,1013],{"className":1012},[583],[585,1014,1015],{"xmlns":587},[589,1016,1017,1023],{},[592,1018,1019,1021],{},[598,1020,886],{},[743,1022,889],{},[605,1024,1025],{"encoding":607},"\\ln",[576,1027,1029],{"className":1028,"ariaHidden":613},[612],[576,1030,1032,1035],{"className":1031},[617],[576,1033],{"className":1034,"style":996},[621],[576,1036,886],{"className":1037},[973]," darf man auch ",[576,1040,1042,1059],{"className":1041},[579],[576,1043,1045],{"className":1044},[583],[585,1046,1047],{"xmlns":587},[589,1048,1049,1056],{},[592,1050,1051,1054],{},[598,1052,1053],{},"lg",[743,1055,889],{},[605,1057,1058],{"encoding":607},"\\lg",[576,1060,1062],{"className":1061,"ariaHidden":613},[612],[576,1063,1065,1068],{"className":1064},[617],[576,1066],{"className":1067,"style":969},[621],[576,1069,1071,1072],{"className":1070},[973],"l",[576,1073,1075],{"style":1074},"margin-right:0.0139em;","g"," oder jeden anderen Logarithmus nehmen, solange oben und unten derselbe steht.",[564,1078,1079,1080,1149],{},"Kennt man ",[576,1081,1083,1100],{"className":1082},[579],[576,1084,1086],{"className":1085},[583],[585,1087,1088],{"xmlns":587},[589,1089,1090,1098],{},[592,1091,1092],{},[595,1093,1094,1096],{},[598,1095,600],{},[598,1097,603],{},[605,1099,608],{"encoding":607},[576,1101,1103],{"className":1102,"ariaHidden":613},[612],[576,1104,1106,1109],{"className":1105},[617],[576,1107],{"className":1108,"style":622},[621],[576,1110,1112,1115],{"className":1111},[626],[576,1113,600],{"className":1114,"style":631},[626,630],[576,1116,1118],{"className":1117},[635],[576,1119,1121,1141],{"className":1120},[639,640],[576,1122,1124,1138],{"className":1123},[644],[576,1125,1127],{"className":1126,"style":649},[648],[576,1128,1129,1132],{"style":652},[576,1130],{"className":1131,"style":657},[656],[576,1133,1135],{"className":1134},[661,662,663,664],[576,1136,603],{"className":1137,"style":668},[626,630,664],[576,1139,673],{"className":1140},[672],[576,1142,1144],{"className":1143},[644],[576,1145,1147],{"className":1146,"style":680},[648],[576,1148],{},", lässt sich der Bestand direkt angeben:",[576,1151,1154],{"className":1152},[1153],"katex-display",[576,1155,1157,1209],{"className":1156},[579],[576,1158,1160],{"className":1159},[583],[585,1161,1163],{"xmlns":587,"display":1162},"block",[589,1164,1165,1206],{},[592,1166,1167,1170,1174,1177,1180,1182,1189,1191],{},[598,1168,1169],{},"N",[743,1171,1173],{"stretchy":1172},"false","(",[598,1175,1176],{},"t",[743,1178,1179],{"stretchy":1172},")",[743,1181,745],{},[595,1183,1184,1186],{},[598,1185,1169],{},[747,1187,1188],{},"0",[743,1190,883],{},[732,1192,1193,1195],{},[747,1194,749],{},[1196,1197,1198,1200],"mfrac",{},[598,1199,1176],{},[595,1201,1202,1204],{},[598,1203,600],{},[598,1205,603],{},[605,1207,1208],{"encoding":607},"N(t) = N_0 \\cdot 2^{\\frac{t}{T_D}}",[576,1210,1212,1243,1300],{"className":1211,"ariaHidden":613},[612],[576,1213,1215,1219,1223,1227,1230,1234,1237,1240],{"className":1214},[617],[576,1216],{"className":1217,"style":1218},[621],"height:1em;vertical-align:-0.25em;",[576,1220,1169],{"className":1221,"style":1222},[626,630],"margin-right:0.109em;",[576,1224,1173],{"className":1225},[1226],"mopen",[576,1228,1176],{"className":1229},[626,630],[576,1231,1179],{"className":1232},[1233],"mclose",[576,1235],{"className":1236,"style":844},[843],[576,1238,745],{"className":1239},[848],[576,1241],{"className":1242,"style":844},[843],[576,1244,1246,1249,1291,1294,1297],{"className":1245},[617],[576,1247],{"className":1248,"style":622},[621],[576,1250,1252,1255],{"className":1251},[626],[576,1253,1169],{"className":1254,"style":1222},[626,630],[576,1256,1258],{"className":1257},[635],[576,1259,1261,1283],{"className":1260},[639,640],[576,1262,1264,1280],{"className":1263},[644],[576,1265,1268],{"className":1266,"style":1267},[648],"height:0.3011em;",[576,1269,1271,1274],{"style":1270},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[576,1272],{"className":1273,"style":657},[656],[576,1275,1277],{"className":1276},[661,662,663,664],[576,1278,1188],{"className":1279},[626,664],[576,1281,673],{"className":1282},[672],[576,1284,1286],{"className":1285},[644],[576,1287,1289],{"className":1288,"style":680},[648],[576,1290],{},[576,1292],{"className":1293,"style":955},[843],[576,1295,883],{"className":1296},[959],[576,1298],{"className":1299,"style":955},[843],[576,1301,1303,1307],{"className":1302},[617],[576,1304],{"className":1305,"style":1306},[621],"height:1.0942em;",[576,1308,1310,1313],{"className":1309},[626],[576,1311,749],{"className":1312},[626],[576,1314,1316],{"className":1315},[635],[576,1317,1319],{"className":1318},[639],[576,1320,1322],{"className":1321},[644],[576,1323,1325],{"className":1324,"style":1306},[648],[576,1326,1328,1332],{"style":1327},"top:-3.5178em;margin-right:0.05em;",[576,1329],{"className":1330,"style":1331},[656],"height:3em;",[576,1333,1335],{"className":1334},[661,662,663,664],[576,1336,1338],{"className":1337},[626,664],[576,1339,1341,1346,1449],{"className":1340},[626,664],[576,1342],{"className":1343},[1226,1344,661,823,1345],"nulldelimiter","size6",[576,1347,1349],{"className":1348},[1196],[576,1350,1352,1440],{"className":1351},[639,640],[576,1353,1355,1437],{"className":1354},[644],[576,1356,1359,1411,1422],{"className":1357,"style":1358},[648],"height:0.8233em;",[576,1360,1362,1365],{"style":1361},"top:-2.656em;",[576,1363],{"className":1364,"style":1331},[656],[576,1366,1368],{"className":1367},[661,823,824,664],[576,1369,1371],{"className":1370},[626,664],[576,1372,1374,1377],{"className":1373},[626,664],[576,1375,600],{"className":1376,"style":631},[626,630,664],[576,1378,1380],{"className":1379},[635],[576,1381,1383,1402],{"className":1382},[639,640],[576,1384,1386,1399],{"className":1385},[644],[576,1387,1389],{"className":1388,"style":812},[648],[576,1390,1392,1396],{"style":1391},"top:-2.3448em;margin-left:-0.1389em;margin-right:0.1em;",[576,1393],{"className":1394,"style":1395},[656],"height:2.6833em;",[576,1397,603],{"className":1398,"style":668},[626,630,664],[576,1400,673],{"className":1401},[672],[576,1403,1405],{"className":1404},[644],[576,1406,1409],{"className":1407,"style":1408},[648],"height:0.3385em;",[576,1410],{},[576,1412,1414,1417],{"style":1413},"top:-3.2255em;",[576,1415],{"className":1416,"style":1331},[656],[576,1418],{"className":1419,"style":1421},[1420,664],"frac-line","border-bottom-width:0.049em;",[576,1423,1425,1428],{"style":1424},"top:-3.384em;",[576,1426],{"className":1427,"style":1331},[656],[576,1429,1431],{"className":1430},[661,823,824,664],[576,1432,1434],{"className":1433},[626,664],[576,1435,1176],{"className":1436},[626,630,664],[576,1438,673],{"className":1439},[672],[576,1441,1443],{"className":1442},[644],[576,1444,1447],{"className":1445,"style":1446},[648],"height:0.5858em;",[576,1448],{},[576,1450],{"className":1451},[1233,1344,661,823,1345],[564,1453,1454,1455,1509],{},"Die Bedingung ",[576,1456,1458,1478],{"className":1457},[579],[576,1459,1461],{"className":1460},[583],[585,1462,1463],{"xmlns":587},[589,1464,1465,1475],{},[592,1466,1467,1469,1472],{},[598,1468,698],{},[743,1470,1471],{},">",[747,1473,1474],{},"1",[605,1476,1477],{"encoding":607},"q > 1",[576,1479,1481,1500],{"className":1480,"ariaHidden":613},[612],[576,1482,1484,1488,1491,1494,1497],{"className":1483},[617],[576,1485],{"className":1486,"style":1487},[621],"height:0.7335em;vertical-align:-0.1944em;",[576,1489,698],{"className":1490,"style":714},[626,630],[576,1492],{"className":1493,"style":844},[843],[576,1495,1471],{"className":1496},[848],[576,1498],{"className":1499,"style":844},[843],[576,1501,1503,1506],{"className":1502},[617],[576,1504],{"className":1505,"style":858},[621],[576,1507,1474],{"className":1508},[626]," gilt, weil nur ein wachsender Bestand sich verdoppelt. Für abnehmende Bestände rechnet man mit der Halbwertszeit.",[564,1511,1512,1513,1516,1517,1687,1688,1716,1717,1866,1867,1942,1943,2035,2036,2088],{},"Als ",[568,1514,1515],{},"Faustregel"," gilt für kleine Prozentsätze ",[576,1518,1520,1548],{"className":1519},[579],[576,1521,1523],{"className":1522},[583],[585,1524,1525],{"xmlns":587},[589,1526,1527,1545],{},[592,1528,1529,1535,1538],{},[595,1530,1531,1533],{},[598,1532,600],{},[598,1534,603],{},[743,1536,1537],{},"≈",[1196,1539,1540,1543],{},[747,1541,1542],{},"70",[598,1544,564],{},[605,1546,1547],{"encoding":607},"T_D \\approx \\frac{70}{p}",[576,1549,1551,1606],{"className":1550,"ariaHidden":613},[612],[576,1552,1554,1557,1597,1600,1603],{"className":1553},[617],[576,1555],{"className":1556,"style":622},[621],[576,1558,1560,1563],{"className":1559},[626],[576,1561,600],{"className":1562,"style":631},[626,630],[576,1564,1566],{"className":1565},[635],[576,1567,1569,1589],{"className":1568},[639,640],[576,1570,1572,1586],{"className":1571},[644],[576,1573,1575],{"className":1574,"style":649},[648],[576,1576,1577,1580],{"style":652},[576,1578],{"className":1579,"style":657},[656],[576,1581,1583],{"className":1582},[661,662,663,664],[576,1584,603],{"className":1585,"style":668},[626,630,664],[576,1587,673],{"className":1588},[672],[576,1590,1592],{"className":1591},[644],[576,1593,1595],{"className":1594,"style":680},[648],[576,1596],{},[576,1598],{"className":1599,"style":844},[843],[576,1601,1537],{"className":1602},[848],[576,1604],{"className":1605,"style":844},[843],[576,1607,1609,1613],{"className":1608},[617],[576,1610],{"className":1611,"style":1612},[621],"height:1.3262em;vertical-align:-0.4811em;",[576,1614,1616,1619,1684],{"className":1615},[626],[576,1617],{"className":1618},[1226,1344],[576,1620,1622],{"className":1621},[1196],[576,1623,1625,1675],{"className":1624},[639,640],[576,1626,1628,1672],{"className":1627},[644],[576,1629,1632,1647,1657],{"className":1630,"style":1631},[648],"height:0.8451em;",[576,1633,1635,1638],{"style":1634},"top:-2.655em;",[576,1636],{"className":1637,"style":1331},[656],[576,1639,1641],{"className":1640},[661,662,663,664],[576,1642,1644],{"className":1643},[626,664],[576,1645,564],{"className":1646},[626,630,664],[576,1648,1650,1653],{"style":1649},"top:-3.23em;",[576,1651],{"className":1652,"style":1331},[656],[576,1654],{"className":1655,"style":1656},[1420],"border-bottom-width:0.04em;",[576,1658,1660,1663],{"style":1659},"top:-3.394em;",[576,1661],{"className":1662,"style":1331},[656],[576,1664,1666],{"className":1665},[661,662,663,664],[576,1667,1669],{"className":1668},[626,664],[576,1670,1542],{"className":1671},[626,664],[576,1673,673],{"className":1674},[672],[576,1676,1678],{"className":1677},[644],[576,1679,1682],{"className":1680,"style":1681},[648],"height:0.4811em;",[576,1683],{},[576,1685],{"className":1686},[1233,1344],", wobei ",[576,1689,1691,1704],{"className":1690},[579],[576,1692,1694],{"className":1693},[583],[585,1695,1696],{"xmlns":587},[589,1697,1698,1702],{},[592,1699,1700],{},[598,1701,564],{},[605,1703,564],{"encoding":607},[576,1705,1707],{"className":1706,"ariaHidden":613},[612],[576,1708,1710,1713],{"className":1709},[617],[576,1711],{"className":1712,"style":710},[621],[576,1714,564],{"className":1715},[626,630]," die Zunahme in Prozent pro Zeitschritt ist (",[576,1718,1720,1748],{"className":1719},[579],[576,1721,1723],{"className":1722},[583],[585,1724,1725],{"xmlns":587},[589,1726,1727,1745],{},[592,1728,1729,1731,1733,1735,1738],{},[598,1730,698],{},[743,1732,745],{},[747,1734,1474],{},[743,1736,1737],{},"+",[1196,1739,1740,1742],{},[598,1741,564],{},[747,1743,1744],{},"100",[605,1746,1747],{"encoding":607},"q = 1 + \\frac{p}{100}",[576,1749,1751,1769,1788],{"className":1750,"ariaHidden":613},[612],[576,1752,1754,1757,1760,1763,1766],{"className":1753},[617],[576,1755],{"className":1756,"style":710},[621],[576,1758,698],{"className":1759,"style":714},[626,630],[576,1761],{"className":1762,"style":844},[843],[576,1764,745],{"className":1765},[848],[576,1767],{"className":1768,"style":844},[843],[576,1770,1772,1776,1779,1782,1785],{"className":1771},[617],[576,1773],{"className":1774,"style":1775},[621],"height:0.7278em;vertical-align:-0.0833em;",[576,1777,1474],{"className":1778},[626],[576,1780],{"className":1781,"style":955},[843],[576,1783,1737],{"className":1784},[959],[576,1786],{"className":1787,"style":955},[843],[576,1789,1791,1795],{"className":1790},[617],[576,1792],{"className":1793,"style":1794},[621],"height:1.0925em;vertical-align:-0.345em;",[576,1796,1798,1801,1863],{"className":1797},[626],[576,1799],{"className":1800},[1226,1344],[576,1802,1804],{"className":1803},[1196],[576,1805,1807,1854],{"className":1806},[639,640],[576,1808,1810,1851],{"className":1809},[644],[576,1811,1814,1828,1836],{"className":1812,"style":1813},[648],"height:0.7475em;",[576,1815,1816,1819],{"style":1634},[576,1817],{"className":1818,"style":1331},[656],[576,1820,1822],{"className":1821},[661,662,663,664],[576,1823,1825],{"className":1824},[626,664],[576,1826,1744],{"className":1827},[626,664],[576,1829,1830,1833],{"style":1649},[576,1831],{"className":1832,"style":1331},[656],[576,1834],{"className":1835,"style":1656},[1420],[576,1837,1839,1842],{"style":1838},"top:-3.4461em;",[576,1840],{"className":1841,"style":1331},[656],[576,1843,1845],{"className":1844},[661,662,663,664],[576,1846,1848],{"className":1847},[626,664],[576,1849,564],{"className":1850},[626,630,664],[576,1852,673],{"className":1853},[672],[576,1855,1857],{"className":1856},[644],[576,1858,1861],{"className":1859,"style":1860},[648],"height:0.345em;",[576,1862],{},[576,1864],{"className":1865},[1233,1344],"). Sie ist nur eine Näherung (wegen ",[576,1868,1870,1893],{"className":1869},[579],[576,1871,1873],{"className":1872},[583],[585,1874,1875],{"xmlns":587},[589,1876,1877,1890],{},[592,1878,1879,1881,1883,1885,1887],{},[598,1880,886],{},[743,1882,889],{},[747,1884,749],{},[743,1886,1537],{},[747,1888,1889],{},"0,69",[605,1891,1892],{"encoding":607},"\\ln 2 \\approx 0{,}69",[576,1894,1896,1920],{"className":1895,"ariaHidden":613},[612],[576,1897,1899,1902,1905,1908,1911,1914,1917],{"className":1898},[617],[576,1900],{"className":1901,"style":996},[621],[576,1903,886],{"className":1904},[973],[576,1906],{"className":1907,"style":977},[843],[576,1909,749],{"className":1910},[626],[576,1912],{"className":1913,"style":844},[843],[576,1915,1537],{"className":1916},[848],[576,1918],{"className":1919,"style":844},[843],[576,1921,1923,1927,1930,1938],{"className":1922},[617],[576,1924],{"className":1925,"style":1926},[621],"height:0.8389em;vertical-align:-0.1944em;",[576,1928,1188],{"className":1929},[626],[576,1931,1933],{"className":1932},[626],[576,1934,1937],{"className":1935},[1936],"mpunct",",",[576,1939,1941],{"className":1940},[626],"69"," und ",[576,1944,1946,1977],{"className":1945},[579],[576,1947,1949],{"className":1948},[583],[585,1950,1951],{"xmlns":587},[589,1952,1953,1974],{},[592,1954,1955,1957,1959,1961,1963,1965,1968,1970,1972],{},[598,1956,886],{},[743,1958,889],{},[743,1960,1173],{"stretchy":1172},[747,1962,1474],{},[743,1964,1737],{},[598,1966,1967],{},"x",[743,1969,1179],{"stretchy":1172},[743,1971,1537],{},[598,1973,1967],{},[605,1975,1976],{"encoding":607},"\\ln(1 + x) \\approx x",[576,1978,1980,2004,2025],{"className":1979,"ariaHidden":613},[612],[576,1981,1983,1986,1989,1992,1995,1998,2001],{"className":1982},[617],[576,1984],{"className":1985,"style":1218},[621],[576,1987,886],{"className":1988},[973],[576,1990,1173],{"className":1991},[1226],[576,1993,1474],{"className":1994},[626],[576,1996],{"className":1997,"style":955},[843],[576,1999,1737],{"className":2000},[959],[576,2002],{"className":2003,"style":955},[843],[576,2005,2007,2010,2013,2016,2019,2022],{"className":2006},[617],[576,2008],{"className":2009,"style":1218},[621],[576,2011,1967],{"className":2012},[626,630],[576,2014,1179],{"className":2015},[1233],[576,2017],{"className":2018,"style":844},[843],[576,2020,1537],{"className":2021},[848],[576,2023],{"className":2024,"style":844},[843],[576,2026,2028,2032],{"className":2027},[617],[576,2029],{"className":2030,"style":2031},[621],"height:0.4306em;",[576,2033,1967],{"className":2034},[626,630],") und taugt bis etwa ",[576,2037,2039,2058],{"className":2038},[579],[576,2040,2042],{"className":2041},[583],[585,2043,2044],{"xmlns":587},[589,2045,2046,2055],{},[592,2047,2048,2050,2052],{},[598,2049,564],{},[743,2051,745],{},[747,2053,2054],{},"10",[605,2056,2057],{"encoding":607},"p = 10",[576,2059,2061,2079],{"className":2060,"ariaHidden":613},[612],[576,2062,2064,2067,2070,2073,2076],{"className":2063},[617],[576,2065],{"className":2066,"style":710},[621],[576,2068,564],{"className":2069},[626,630],[576,2071],{"className":2072,"style":844},[843],[576,2074,745],{"className":2075},[848],[576,2077],{"className":2078,"style":844},[843],[576,2080,2082,2085],{"className":2081},[617],[576,2083],{"className":2084,"style":858},[621],[576,2086,2054],{"className":2087},[626],"; bei großen Prozentsätzen wird sie ungenau.",[559,2090,2092],{"id":2091},"beispiele","Beispiele",[564,2094,2095,2096,2141,2142,2204],{},"Ein Guthaben wächst um ",[576,2097,2099,2122],{"className":2098},[579],[576,2100,2102],{"className":2101},[583],[585,2103,2104],{"xmlns":587},[589,2105,2106,2119],{},[592,2107,2108,2111,2115],{},[747,2109,2110],{},"5",[2112,2113,2114],"mtext",{}," ",[598,2116,2118],{"mathvariant":2117},"normal","%",[605,2120,2121],{"encoding":607},"5\\,\\%",[576,2123,2125],{"className":2124,"ariaHidden":613},[612],[576,2126,2128,2132,2135,2138],{"className":2127},[617],[576,2129],{"className":2130,"style":2131},[621],"height:0.8056em;vertical-align:-0.0556em;",[576,2133,2110],{"className":2134},[626],[576,2136],{"className":2137,"style":977},[843],[576,2139,2118],{"className":2140},[626]," pro Jahr, also ",[576,2143,2145,2164],{"className":2144},[579],[576,2146,2148],{"className":2147},[583],[585,2149,2150],{"xmlns":587},[589,2151,2152,2161],{},[592,2153,2154,2156,2158],{},[598,2155,698],{},[743,2157,745],{},[747,2159,2160],{},"1,05",[605,2162,2163],{"encoding":607},"q = 1{,}05",[576,2165,2167,2185],{"className":2166,"ariaHidden":613},[612],[576,2168,2170,2173,2176,2179,2182],{"className":2169},[617],[576,2171],{"className":2172,"style":710},[621],[576,2174,698],{"className":2175,"style":714},[626,630],[576,2177],{"className":2178,"style":844},[843],[576,2180,745],{"className":2181},[848],[576,2183],{"className":2184,"style":844},[843],[576,2186,2188,2191,2194,2200],{"className":2187},[617],[576,2189],{"className":2190,"style":1926},[621],[576,2192,1474],{"className":2193},[626],[576,2195,2197],{"className":2196},[626],[576,2198,1937],{"className":2199},[1936],[576,2201,2203],{"className":2202},[626],"05",":",[576,2206,2208],{"className":2207},[1153],[576,2209,2211,2257],{"className":2210},[579],[576,2212,2214],{"className":2213},[583],[585,2215,2216],{"xmlns":587,"display":1162},[589,2217,2218,2254],{},[592,2219,2220,2226,2228,2246,2248,2251],{},[595,2221,2222,2224],{},[598,2223,600],{},[598,2225,603],{},[743,2227,745],{},[1196,2229,2230,2238],{},[592,2231,2232,2234,2236],{},[598,2233,886],{},[743,2235,889],{},[747,2237,749],{},[592,2239,2240,2242,2244],{},[598,2241,886],{},[743,2243,889],{},[747,2245,2160],{},[743,2247,1537],{},[747,2249,2250],{},"14,2",[2112,2252,2253],{}," Jahre",[605,2255,2256],{"encoding":607},"T_D = \\frac{\\ln 2}{\\ln 1{,}05} \\approx 14{,}2 \\text{ Jahre}",[576,2258,2260,2315,2418],{"className":2259,"ariaHidden":613},[612],[576,2261,2263,2266,2306,2309,2312],{"className":2262},[617],[576,2264],{"className":2265,"style":622},[621],[576,2267,2269,2272],{"className":2268},[626],[576,2270,600],{"className":2271,"style":631},[626,630],[576,2273,2275],{"className":2274},[635],[576,2276,2278,2298],{"className":2277},[639,640],[576,2279,2281,2295],{"className":2280},[644],[576,2282,2284],{"className":2283,"style":649},[648],[576,2285,2286,2289],{"style":652},[576,2287],{"className":2288,"style":657},[656],[576,2290,2292],{"className":2291},[661,662,663,664],[576,2293,603],{"className":2294,"style":668},[626,630,664],[576,2296,673],{"className":2297},[672],[576,2299,2301],{"className":2300},[644],[576,2302,2304],{"className":2303,"style":680},[648],[576,2305],{},[576,2307],{"className":2308,"style":844},[843],[576,2310,745],{"className":2311},[848],[576,2313],{"className":2314,"style":844},[843],[576,2316,2318,2322,2409,2412,2415],{"className":2317},[617],[576,2319],{"className":2320,"style":2321},[621],"height:2.2519em;vertical-align:-0.8804em;",[576,2323,2325,2328,2406],{"className":2324},[626],[576,2326],{"className":2327},[1226,1344],[576,2329,2331],{"className":2330},[1196],[576,2332,2334,2397],{"className":2333},[639,640],[576,2335,2337,2394],{"className":2336},[644],[576,2338,2341,2368,2376],{"className":2339,"style":2340},[648],"height:1.3714em;",[576,2342,2344,2347],{"style":2343},"top:-2.314em;",[576,2345],{"className":2346,"style":1331},[656],[576,2348,2350,2353,2356,2359,2365],{"className":2349},[626],[576,2351,886],{"className":2352},[973],[576,2354],{"className":2355,"style":977},[843],[576,2357,1474],{"className":2358},[626],[576,2360,2362],{"className":2361},[626],[576,2363,1937],{"className":2364},[1936],[576,2366,2203],{"className":2367},[626],[576,2369,2370,2373],{"style":1649},[576,2371],{"className":2372,"style":1331},[656],[576,2374],{"className":2375,"style":1656},[1420],[576,2377,2379,2382],{"style":2378},"top:-3.677em;",[576,2380],{"className":2381,"style":1331},[656],[576,2383,2385,2388,2391],{"className":2384},[626],[576,2386,886],{"className":2387},[973],[576,2389],{"className":2390,"style":977},[843],[576,2392,749],{"className":2393},[626],[576,2395,673],{"className":2396},[672],[576,2398,2400],{"className":2399},[644],[576,2401,2404],{"className":2402,"style":2403},[648],"height:0.8804em;",[576,2405],{},[576,2407],{"className":2408},[1233,1344],[576,2410],{"className":2411,"style":844},[843],[576,2413,1537],{"className":2414},[848],[576,2416],{"className":2417,"style":844},[843],[576,2419,2421,2424,2428,2434,2437],{"className":2420},[617],[576,2422],{"className":2423,"style":969},[621],[576,2425,2427],{"className":2426},[626],"14",[576,2429,2431],{"className":2430},[626],[576,2432,1937],{"className":2433},[1936],[576,2435,749],{"className":2436},[626],[576,2438,2441],{"className":2439},[626,2440],"text",[576,2442,2253],{"className":2443},[626],[564,2445,2446,2447,2568],{},"Die Faustregel liefert ",[576,2448,2450,2472],{"className":2449},[579],[576,2451,2453],{"className":2452},[583],[585,2454,2455],{"xmlns":587},[589,2456,2457,2469],{},[592,2458,2459,2465,2467],{},[1196,2460,2461,2463],{},[747,2462,1542],{},[747,2464,2110],{},[743,2466,745],{},[747,2468,2427],{},[605,2470,2471],{"encoding":607},"\\frac{70}{5} = 14",[576,2473,2475,2559],{"className":2474,"ariaHidden":613},[612],[576,2476,2478,2482,2550,2553,2556],{"className":2477},[617],[576,2479],{"className":2480,"style":2481},[621],"height:1.1901em;vertical-align:-0.345em;",[576,2483,2485,2488,2547],{"className":2484},[626],[576,2486],{"className":2487},[1226,1344],[576,2489,2491],{"className":2490},[1196],[576,2492,2494,2539],{"className":2493},[639,640],[576,2495,2497,2536],{"className":2496},[644],[576,2498,2500,2514,2522],{"className":2499,"style":1631},[648],[576,2501,2502,2505],{"style":1634},[576,2503],{"className":2504,"style":1331},[656],[576,2506,2508],{"className":2507},[661,662,663,664],[576,2509,2511],{"className":2510},[626,664],[576,2512,2110],{"className":2513},[626,664],[576,2515,2516,2519],{"style":1649},[576,2517],{"className":2518,"style":1331},[656],[576,2520],{"className":2521,"style":1656},[1420],[576,2523,2524,2527],{"style":1659},[576,2525],{"className":2526,"style":1331},[656],[576,2528,2530],{"className":2529},[661,662,663,664],[576,2531,2533],{"className":2532},[626,664],[576,2534,1542],{"className":2535},[626,664],[576,2537,673],{"className":2538},[672],[576,2540,2542],{"className":2541},[644],[576,2543,2545],{"className":2544,"style":1860},[648],[576,2546],{},[576,2548],{"className":2549},[1233,1344],[576,2551],{"className":2552,"style":844},[843],[576,2554,745],{"className":2555},[848],[576,2557],{"className":2558,"style":844},[843],[576,2560,2562,2565],{"className":2561},[617],[576,2563],{"className":2564,"style":858},[621],[576,2566,2427],{"className":2567},[626]," Jahre, also fast dasselbe.",[564,2570,2571,2572,2600,2601,2630,2631,2659,2660,2689],{},"Eine Bakterienkultur mit ",[576,2573,2575,2588],{"className":2574},[579],[576,2576,2578],{"className":2577},[583],[585,2579,2580],{"xmlns":587},[589,2581,2582,2586],{},[592,2583,2584],{},[747,2585,1744],{},[605,2587,1744],{"encoding":607},[576,2589,2591],{"className":2590,"ariaHidden":613},[612],[576,2592,2594,2597],{"className":2593},[617],[576,2595],{"className":2596,"style":858},[621],[576,2598,1744],{"className":2599},[626]," Bakterien verdoppelt sich alle ",[576,2602,2604,2618],{"className":2603},[579],[576,2605,2607],{"className":2606},[583],[585,2608,2609],{"xmlns":587},[589,2610,2611,2616],{},[592,2612,2613],{},[747,2614,2615],{},"20",[605,2617,2615],{"encoding":607},[576,2619,2621],{"className":2620,"ariaHidden":613},[612],[576,2622,2624,2627],{"className":2623},[617],[576,2625],{"className":2626,"style":858},[621],[576,2628,2615],{"className":2629},[626]," Minuten. Nach ",[576,2632,2634,2647],{"className":2633},[579],[576,2635,2637],{"className":2636},[583],[585,2638,2639],{"xmlns":587},[589,2640,2641,2645],{},[592,2642,2643],{},[747,2644,749],{},[605,2646,749],{"encoding":607},[576,2648,2650],{"className":2649,"ariaHidden":613},[612],[576,2651,2653,2656],{"className":2652},[617],[576,2654],{"className":2655,"style":858},[621],[576,2657,749],{"className":2658},[626]," Stunden, also ",[576,2661,2663,2677],{"className":2662},[579],[576,2664,2666],{"className":2665},[583],[585,2667,2668],{"xmlns":587},[589,2669,2670,2675],{},[592,2671,2672],{},[747,2673,2674],{},"120",[605,2676,2674],{"encoding":607},[576,2678,2680],{"className":2679,"ariaHidden":613},[612],[576,2681,2683,2686],{"className":2682},[617],[576,2684],{"className":2685,"style":858},[621],[576,2687,2674],{"className":2688},[626]," Minuten, sind es",[576,2691,2693],{"className":2692},[1153],[576,2694,2696,2750],{"className":2695},[579],[576,2697,2699],{"className":2698},[583],[585,2700,2701],{"xmlns":587,"display":1162},[589,2702,2703,2747],{},[592,2704,2705,2707,2709,2711,2713,2715,2717,2719,2729,2731,2733,2735,2742,2744],{},[598,2706,1169],{},[743,2708,1173],{"stretchy":1172},[747,2710,2674],{},[743,2712,1179],{"stretchy":1172},[743,2714,745],{},[747,2716,1744],{},[743,2718,883],{},[732,2720,2721,2723],{},[747,2722,749],{},[1196,2724,2725,2727],{},[747,2726,2674],{},[747,2728,2615],{},[743,2730,745],{},[747,2732,1744],{},[743,2734,883],{},[732,2736,2737,2739],{},[747,2738,749],{},[747,2740,2741],{},"6",[743,2743,745],{},[747,2745,2746],{},"6400",[605,2748,2749],{"encoding":607},"N(120) = 100 \\cdot 2^{\\frac{120}{20}} = 100 \\cdot 2^6 = 6400",[576,2751,2753,2780,2798,2914,2932,2978],{"className":2752,"ariaHidden":613},[612],[576,2754,2756,2759,2762,2765,2768,2771,2774,2777],{"className":2755},[617],[576,2757],{"className":2758,"style":1218},[621],[576,2760,1169],{"className":2761,"style":1222},[626,630],[576,2763,1173],{"className":2764},[1226],[576,2766,2674],{"className":2767},[626],[576,2769,1179],{"className":2770},[1233],[576,2772],{"className":2773,"style":844},[843],[576,2775,745],{"className":2776},[848],[576,2778],{"className":2779,"style":844},[843],[576,2781,2783,2786,2789,2792,2795],{"className":2782},[617],[576,2784],{"className":2785,"style":858},[621],[576,2787,1744],{"className":2788},[626],[576,2790],{"className":2791,"style":955},[843],[576,2793,883],{"className":2794},[959],[576,2796],{"className":2797,"style":955},[843],[576,2799,2801,2805,2905,2908,2911],{"className":2800},[617],[576,2802],{"className":2803,"style":2804},[621],"height:1.004em;",[576,2806,2808,2811],{"className":2807},[626],[576,2809,749],{"className":2810},[626],[576,2812,2814],{"className":2813},[635],[576,2815,2817],{"className":2816},[639],[576,2818,2820],{"className":2819},[644],[576,2821,2823],{"className":2822,"style":2804},[648],[576,2824,2826,2829],{"style":2825},"top:-3.413em;margin-right:0.05em;",[576,2827],{"className":2828,"style":1331},[656],[576,2830,2832],{"className":2831},[661,662,663,664],[576,2833,2835],{"className":2834},[626,664],[576,2836,2838,2841,2902],{"className":2837},[626,664],[576,2839],{"className":2840},[1226,1344,661,823,1345],[576,2842,2844],{"className":2843},[1196],[576,2845,2847,2893],{"className":2846},[639,640],[576,2848,2850,2890],{"className":2849},[644],[576,2851,2854,2868,2876],{"className":2852,"style":2853},[648],"height:0.8443em;",[576,2855,2856,2859],{"style":1361},[576,2857],{"className":2858,"style":1331},[656],[576,2860,2862],{"className":2861},[661,823,824,664],[576,2863,2865],{"className":2864},[626,664],[576,2866,2615],{"className":2867},[626,664],[576,2869,2870,2873],{"style":1413},[576,2871],{"className":2872,"style":1331},[656],[576,2874],{"className":2875,"style":1421},[1420,664],[576,2877,2878,2881],{"style":1424},[576,2879],{"className":2880,"style":1331},[656],[576,2882,2884],{"className":2883},[661,823,824,664],[576,2885,2887],{"className":2886},[626,664],[576,2888,2674],{"className":2889},[626,664],[576,2891,673],{"className":2892},[672],[576,2894,2896],{"className":2895},[644],[576,2897,2900],{"className":2898,"style":2899},[648],"height:0.344em;",[576,2901],{},[576,2903],{"className":2904},[1233,1344,661,823,1345],[576,2906],{"className":2907,"style":844},[843],[576,2909,745],{"className":2910},[848],[576,2912],{"className":2913,"style":844},[843],[576,2915,2917,2920,2923,2926,2929],{"className":2916},[617],[576,2918],{"className":2919,"style":858},[621],[576,2921,1744],{"className":2922},[626],[576,2924],{"className":2925,"style":955},[843],[576,2927,883],{"className":2928},[959],[576,2930],{"className":2931,"style":955},[843],[576,2933,2935,2939,2969,2972,2975],{"className":2934},[617],[576,2936],{"className":2937,"style":2938},[621],"height:0.8641em;",[576,2940,2942,2945],{"className":2941},[626],[576,2943,749],{"className":2944},[626],[576,2946,2948],{"className":2947},[635],[576,2949,2951],{"className":2950},[639],[576,2952,2954],{"className":2953},[644],[576,2955,2957],{"className":2956,"style":2938},[648],[576,2958,2960,2963],{"style":2959},"top:-3.113em;margin-right:0.05em;",[576,2961],{"className":2962,"style":657},[656],[576,2964,2966],{"className":2965},[661,662,663,664],[576,2967,2741],{"className":2968},[626,664],[576,2970],{"className":2971,"style":844},[843],[576,2973,745],{"className":2974},[848],[576,2976],{"className":2977,"style":844},[843],[576,2979,2981,2984],{"className":2980},[617],[576,2982],{"className":2983,"style":858},[621],[576,2985,2746],{"className":2986},[626],[559,2988,2990],{"id":2989},"typische-fehler","Typische Fehler",[2992,2993,2994,3208,3482],"ul",{},[2995,2996,2997,2998,3096,3097,3135,3136,3164,3165,3203,3204,3207],"li",{},"Mit ",[576,2999,3001,3019],{"className":3000},[579],[576,3002,3004],{"className":3003},[583],[585,3005,3006],{"xmlns":587},[589,3007,3008,3016],{},[592,3009,3010],{},[1196,3011,3012,3014],{},[747,3013,1744],{},[598,3015,564],{},[605,3017,3018],{"encoding":607},"\\frac{100}{p}",[576,3020,3022],{"className":3021,"ariaHidden":613},[612],[576,3023,3025,3028],{"className":3024},[617],[576,3026],{"className":3027,"style":1612},[621],[576,3029,3031,3034,3093],{"className":3030},[626],[576,3032],{"className":3033},[1226,1344],[576,3035,3037],{"className":3036},[1196],[576,3038,3040,3085],{"className":3039},[639,640],[576,3041,3043,3082],{"className":3042},[644],[576,3044,3046,3060,3068],{"className":3045,"style":1631},[648],[576,3047,3048,3051],{"style":1634},[576,3049],{"className":3050,"style":1331},[656],[576,3052,3054],{"className":3053},[661,662,663,664],[576,3055,3057],{"className":3056},[626,664],[576,3058,564],{"className":3059},[626,630,664],[576,3061,3062,3065],{"style":1649},[576,3063],{"className":3064,"style":1331},[656],[576,3066],{"className":3067,"style":1656},[1420],[576,3069,3070,3073],{"style":1659},[576,3071],{"className":3072,"style":1331},[656],[576,3074,3076],{"className":3075},[661,662,663,664],[576,3077,3079],{"className":3078},[626,664],[576,3080,1744],{"className":3081},[626,664],[576,3083,673],{"className":3084},[672],[576,3086,3088],{"className":3087},[644],[576,3089,3091],{"className":3090,"style":1681},[648],[576,3092],{},[576,3094],{"className":3095},[1233,1344]," rechnen: Bei ",[576,3098,3100,3117],{"className":3099},[579],[576,3101,3103],{"className":3102},[583],[585,3104,3105],{"xmlns":587},[589,3106,3107,3115],{},[592,3108,3109,3111,3113],{},[747,3110,2110],{},[2112,3112,2114],{},[598,3114,2118],{"mathvariant":2117},[605,3116,2121],{"encoding":607},[576,3118,3120],{"className":3119,"ariaHidden":613},[612],[576,3121,3123,3126,3129,3132],{"className":3122},[617],[576,3124],{"className":3125,"style":2131},[621],[576,3127,2110],{"className":3128},[626],[576,3130],{"className":3131,"style":977},[843],[576,3133,2118],{"className":3134},[626]," wären das ",[576,3137,3139,3152],{"className":3138},[579],[576,3140,3142],{"className":3141},[583],[585,3143,3144],{"xmlns":587},[589,3145,3146,3150],{},[592,3147,3148],{},[747,3149,2615],{},[605,3151,2615],{"encoding":607},[576,3153,3155],{"className":3154,"ariaHidden":613},[612],[576,3156,3158,3161],{"className":3157},[617],[576,3159],{"className":3160,"style":858},[621],[576,3162,2615],{"className":3163},[626]," Jahre. Das gilt nur bei linearem Wachstum, bei dem jedes Jahr ",[576,3166,3168,3185],{"className":3167},[579],[576,3169,3171],{"className":3170},[583],[585,3172,3173],{"xmlns":587},[589,3174,3175,3183],{},[592,3176,3177,3179,3181],{},[747,3178,2110],{},[2112,3180,2114],{},[598,3182,2118],{"mathvariant":2117},[605,3184,2121],{"encoding":607},[576,3186,3188],{"className":3187,"ariaHidden":613},[612],[576,3189,3191,3194,3197,3200],{"className":3190},[617],[576,3192],{"className":3193,"style":2131},[621],[576,3195,2110],{"className":3196},[626],[576,3198],{"className":3199,"style":977},[843],[576,3201,2118],{"className":3202},[626]," des ",[568,3205,3206],{},"Anfangswertes"," dazukommen.",[2995,3209,3210,3211,3382,3383,3421,3422,3481],{},"Den Prozentsatz statt des Faktors einsetzen: ",[576,3212,3214,3249],{"className":3213},[579],[576,3215,3217],{"className":3216},[583],[585,3218,3219],{"xmlns":587},[589,3220,3221,3246],{},[592,3222,3223,3241,3243],{},[1196,3224,3225,3233],{},[592,3226,3227,3229,3231],{},[598,3228,886],{},[743,3230,889],{},[747,3232,749],{},[592,3234,3235,3237,3239],{},[598,3236,886],{},[743,3238,889],{},[747,3240,2110],{},[743,3242,1537],{},[747,3244,3245],{},"0,43",[605,3247,3248],{"encoding":607},"\\frac{\\ln 2}{\\ln 5} \\approx 0{,}43",[576,3250,3252,3363],{"className":3251,"ariaHidden":613},[612],[576,3253,3255,3259,3354,3357,3360],{"className":3254},[617],[576,3256],{"className":3257,"style":3258},[621],"height:1.2251em;vertical-align:-0.345em;",[576,3260,3262,3265,3351],{"className":3261},[626],[576,3263],{"className":3264},[1226,1344],[576,3266,3268],{"className":3267},[1196],[576,3269,3271,3343],{"className":3270},[639,640],[576,3272,3274,3340],{"className":3273},[644],[576,3275,3278,3306,3314],{"className":3276,"style":3277},[648],"height:0.8801em;",[576,3279,3280,3283],{"style":1634},[576,3281],{"className":3282,"style":1331},[656],[576,3284,3286],{"className":3285},[661,662,663,664],[576,3287,3289,3299,3303],{"className":3288},[626,664],[576,3290,3292,3295],{"className":3291},[973,664],[576,3293,1071],{"className":3294},[664],[576,3296,3298],{"className":3297},[664],"n",[576,3300],{"className":3301,"style":3302},[843,664],"margin-right:0.1952em;",[576,3304,2110],{"className":3305},[626,664],[576,3307,3308,3311],{"style":1649},[576,3309],{"className":3310,"style":1331},[656],[576,3312],{"className":3313,"style":1656},[1420],[576,3315,3316,3319],{"style":1659},[576,3317],{"className":3318,"style":1331},[656],[576,3320,3322],{"className":3321},[661,662,663,664],[576,3323,3325,3334,3337],{"className":3324},[626,664],[576,3326,3328,3331],{"className":3327},[973,664],[576,3329,1071],{"className":3330},[664],[576,3332,3298],{"className":3333},[664],[576,3335],{"className":3336,"style":3302},[843,664],[576,3338,749],{"className":3339},[626,664],[576,3341,673],{"className":3342},[672],[576,3344,3346],{"className":3345},[644],[576,3347,3349],{"className":3348,"style":1860},[648],[576,3350],{},[576,3352],{"className":3353},[1233,1344],[576,3355],{"className":3356,"style":844},[843],[576,3358,1537],{"className":3359},[848],[576,3361],{"className":3362,"style":844},[843],[576,3364,3366,3369,3372,3378],{"className":3365},[617],[576,3367],{"className":3368,"style":1926},[621],[576,3370,1188],{"className":3371},[626],[576,3373,3375],{"className":3374},[626],[576,3376,1937],{"className":3377},[1936],[576,3379,3381],{"className":3380},[626],"43"," ist falsch. Bei ",[576,3384,3386,3403],{"className":3385},[579],[576,3387,3389],{"className":3388},[583],[585,3390,3391],{"xmlns":587},[589,3392,3393,3401],{},[592,3394,3395,3397,3399],{},[747,3396,2110],{},[2112,3398,2114],{},[598,3400,2118],{"mathvariant":2117},[605,3402,2121],{"encoding":607},[576,3404,3406],{"className":3405,"ariaHidden":613},[612],[576,3407,3409,3412,3415,3418],{"className":3408},[617],[576,3410],{"className":3411,"style":2131},[621],[576,3413,2110],{"className":3414},[626],[576,3416],{"className":3417,"style":977},[843],[576,3419,2118],{"className":3420},[626]," ist ",[576,3423,3425,3442],{"className":3424},[579],[576,3426,3428],{"className":3427},[583],[585,3429,3430],{"xmlns":587},[589,3431,3432,3440],{},[592,3433,3434,3436,3438],{},[598,3435,698],{},[743,3437,745],{},[747,3439,2160],{},[605,3441,2163],{"encoding":607},[576,3443,3445,3463],{"className":3444,"ariaHidden":613},[612],[576,3446,3448,3451,3454,3457,3460],{"className":3447},[617],[576,3449],{"className":3450,"style":710},[621],[576,3452,698],{"className":3453,"style":714},[626,630],[576,3455],{"className":3456,"style":844},[843],[576,3458,745],{"className":3459},[848],[576,3461],{"className":3462,"style":844},[843],[576,3464,3466,3469,3472,3478],{"className":3465},[617],[576,3467],{"className":3468,"style":1926},[621],[576,3470,1474],{"className":3471},[626],[576,3473,3475],{"className":3474},[626],[576,3476,1937],{"className":3477},[1936],[576,3479,2203],{"className":3480},[626],".",[2995,3483,3484,3485,3554,3555,3583],{},"Zeiteinheiten mischen: ",[576,3486,3488,3505],{"className":3487},[579],[576,3489,3491],{"className":3490},[583],[585,3492,3493],{"xmlns":587},[589,3494,3495,3503],{},[592,3496,3497],{},[595,3498,3499,3501],{},[598,3500,600],{},[598,3502,603],{},[605,3504,608],{"encoding":607},[576,3506,3508],{"className":3507,"ariaHidden":613},[612],[576,3509,3511,3514],{"className":3510},[617],[576,3512],{"className":3513,"style":622},[621],[576,3515,3517,3520],{"className":3516},[626],[576,3518,600],{"className":3519,"style":631},[626,630],[576,3521,3523],{"className":3522},[635],[576,3524,3526,3546],{"className":3525},[639,640],[576,3527,3529,3543],{"className":3528},[644],[576,3530,3532],{"className":3531,"style":649},[648],[576,3533,3534,3537],{"style":652},[576,3535],{"className":3536,"style":657},[656],[576,3538,3540],{"className":3539},[661,662,663,664],[576,3541,603],{"className":3542,"style":668},[626,630,664],[576,3544,673],{"className":3545},[672],[576,3547,3549],{"className":3548},[644],[576,3550,3552],{"className":3551,"style":680},[648],[576,3553],{}," kommt in der Einheit heraus, auf die sich ",[576,3556,3558,3571],{"className":3557},[579],[576,3559,3561],{"className":3560},[583],[585,3562,3563],{"xmlns":587},[589,3564,3565,3569],{},[592,3566,3567],{},[598,3568,698],{},[605,3570,698],{"encoding":607},[576,3572,3574],{"className":3573,"ariaHidden":613},[612],[576,3575,3577,3580],{"className":3576},[617],[576,3578],{"className":3579,"style":710},[621],[576,3581,698],{"className":3582,"style":714},[626,630]," bezieht.",{"title":3585,"searchDepth":3586,"depth":3586,"links":3587},"",2,[3588,3589,3590],{"id":561,"depth":3586,"text":562},{"id":2091,"depth":3586,"text":2092},{"id":2989,"depth":3586,"text":2990},[3592,3594,3597,3600,3604,3607,3609,3615,3617],{"lhs":3593,"rhs":749},"q^(log(2)\u002Flog(q))",{"lhs":3595,"rhs":3596},"log(2)\u002Flog(q)","log10(2)\u002Flog10(q)",{"lhs":3598,"rhs":3599},"q^(2*log(2)\u002Flog(q))","4",{"lhs":3595,"rhs":3601,"where":3602},"log(2)\u002Flog(1.05)",{"q":3603},1.05,{"lhs":3605,"rhs":3606},"round(log(2)\u002Flog(1.05), 1)","14.2",{"lhs":3608,"rhs":2427},"70\u002F5",{"lhs":3610,"rhs":2746,"where":3611},"N0*2^(t\u002FT)",{"N0":3612,"t":3613,"T":3614},100,120,20,{"lhs":3616,"rhs":2615},"100\u002F5",{"lhs":3618,"rhs":3619},"round(log(2)\u002Flog(5), 2)","0.43","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>q\u003C\u002Fmi>\u003Cmo>>\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">q > 1\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.7335em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">q\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md",{"caption":3624,"x":3625,"y":3628,"unit":3631,"unitY":3632,"ticks":3633,"axes":3635,"items":3636},"Graph von N(t) = 100 · 2^(t\u002F20); nach jeder Verdopplungszeit T_D = 20 Minuten hat sich der Bestand verdoppelt",[3626,3627],-12,68,[3629,3630],-80,1000,3.6,0.24,[3612,3634],200,true,[3637,3643,3648,3653,3656,3659,3662,3666,3673,3675,3677,3679,3683,3686],{"type":3638,"from":3639,"to":3641,"style":3642},"segment",[3614,3640],0,[3614,3634],"dashed",{"type":3638,"from":3644,"to":3646,"style":3642},[3645,3640],40,[3645,3647],400,{"type":3638,"from":3649,"to":3651,"style":3642},[3650,3640],60,[3650,3652],800,{"type":3638,"from":3654,"to":3655,"style":3642},[3640,3634],[3614,3634],{"type":3638,"from":3657,"to":3658,"style":3642},[3640,3647],[3645,3647],{"type":3638,"from":3660,"to":3661,"style":3642},[3640,3652],[3650,3652],{"type":3663,"expr":3664,"domain":3665},"function","100*2^(x\u002F20)",[3640,3627],{"type":3667,"at":3668,"label":3669,"offset":3670},"point",[3640,3612],"N_0",[3671,3672],8,-8,{"type":3667,"at":3674},[3614,3634],{"type":3667,"at":3676},[3645,3647],{"type":3667,"at":3678},[3650,3652],{"type":3680,"at":3681,"text":608},"label",[3614,3682],-45,{"type":3680,"at":3684,"text":3685},[3645,3682],"2T_D",{"type":3680,"at":3687,"text":3688},[3650,3682],"3T_D","\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"0 0 288 259.2\" width=\"288\" height=\"259.2\" role=\"img\" aria-label=\"Graph von N(t) = 100 · 2^(t\u002F20); nach jeder Verdopplungszeit T_D = 20 Minuten hat sich der Bestand verdoppelt\" class=\"formula-figure-svg\">\u003Cdefs>\u003Cmarker id=\"fig-doubling-time-arrow\" viewBox=\"0 0 10 10\" refX=\"9\" refY=\"5\" markerWidth=\"7\" markerHeight=\"7\" orient=\"auto-start-reverse\">\u003Cpath d=\"M 0 0 L 10 5 L 0 10 z\" class=\"fig-arrowhead\"\u002F>\u003C\u002Fmarker>\u003CclipPath id=\"fig-doubling-time-clip\">\u003Crect x=\"0\" y=\"0\" width=\"288\" height=\"259.2\"\u002F>\u003C\u002FclipPath>\u003C\u002Fdefs>\u003Cline x1=\"0\" y1=\"240\" x2=\"288\" y2=\"240\" class=\"fig-axis\" marker-end=\"url(#fig-doubling-time-arrow)\"\u002F>\u003Cline x1=\"43.2\" y1=\"259.2\" x2=\"43.2\" y2=\"0\" class=\"fig-axis\" marker-end=\"url(#fig-doubling-time-arrow)\"\u002F>\u003Cline x1=\"40.2\" y1=\"192\" x2=\"46.2\" y2=\"192\" class=\"fig-axis\"\u002F>\u003Ctext x=\"37.2\" y=\"185\" text-anchor=\"end\" dominant-baseline=\"middle\" class=\"fig-tick\">200\u003C\u002Ftext>\u003Cline x1=\"40.2\" y1=\"144\" x2=\"46.2\" y2=\"144\" class=\"fig-axis\"\u002F>\u003Ctext x=\"37.2\" y=\"137\" text-anchor=\"end\" dominant-baseline=\"middle\" class=\"fig-tick\">400\u003C\u002Ftext>\u003Cline x1=\"40.2\" y1=\"96\" x2=\"46.2\" y2=\"96\" class=\"fig-axis\"\u002F>\u003Ctext x=\"37.2\" y=\"89\" text-anchor=\"end\" dominant-baseline=\"middle\" class=\"fig-tick\">600\u003C\u002Ftext>\u003Cline x1=\"40.2\" y1=\"48\" x2=\"46.2\" y2=\"48\" class=\"fig-axis\"\u002F>\u003Ctext x=\"37.2\" y=\"41\" text-anchor=\"end\" dominant-baseline=\"middle\" class=\"fig-tick\">800\u003C\u002Ftext>\u003Ctext x=\"282\" y=\"228\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">x\u003C\u002Ftext>\u003Ctext x=\"55.2\" y=\"8\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">y\u003C\u002Ftext>\u003Cline x1=\"115.2\" y1=\"240\" x2=\"115.2\" y2=\"192\" class=\"fig-dashed\"\u002F>\u003Cline x1=\"187.2\" y1=\"240\" x2=\"187.2\" y2=\"144\" class=\"fig-dashed\"\u002F>\u003Cline x1=\"259.2\" y1=\"240\" x2=\"259.2\" y2=\"48\" class=\"fig-dashed\"\u002F>\u003Cline x1=\"43.2\" y1=\"192\" x2=\"115.2\" y2=\"192\" class=\"fig-dashed\"\u002F>\u003Cline x1=\"43.2\" y1=\"144\" x2=\"187.2\" y2=\"144\" class=\"fig-dashed\"\u002F>\u003Cline x1=\"43.2\" y1=\"48\" x2=\"259.2\" y2=\"48\" class=\"fig-dashed\"\u002F>\u003Cpath d=\"M 43.2 216 L 44.22 215.76 L 45.24 215.52 L 46.26 215.28 L 47.28 215.04 L 48.3 214.79 L 49.32 214.54 L 50.34 214.29 L 51.36 214.04 L 52.38 213.78 L 53.4 213.52 L 54.42 213.26 L 55.44 213 L 56.46 212.73 L 57.48 212.46 L 58.5 212.19 L 59.52 211.92 L 60.54 211.64 L 61.56 211.36 L 62.58 211.08 L 63.6 210.79 L 64.62 210.5 L 65.64 210.21 L 66.66 209.92 L 67.68 209.62 L 68.7 209.32 L 69.72 209.02 L 70.74 208.71 L 71.76 208.4 L 72.78 208.09 L 73.8 207.78 L 74.82 207.46 L 75.84 207.14 L 76.86 206.81 L 77.88 206.49 L 78.9 206.16 L 79.92 205.82 L 80.94 205.49 L 81.96 205.14 L 82.98 204.8 L 84 204.45 L 85.02 204.1 L 86.04 203.75 L 87.06 203.39 L 88.08 203.03 L 89.1 202.66 L 90.12 202.3 L 91.14 201.92 L 92.16 201.55 L 93.18 201.17 L 94.2 200.79 L 95.22 200.4 L 96.24 200.01 L 97.26 199.61 L 98.28 199.22 L 99.3 198.81 L 100.32 198.41 L 101.34 198 L 102.36 197.58 L 103.38 197.16 L 104.4 196.74 L 105.42 196.31 L 106.44 195.88 L 107.46 195.45 L 108.48 195.01 L 109.5 194.56 L 110.52 194.11 L 111.54 193.66 L 112.56 193.2 L 113.58 192.74 L 114.6 192.28 L 115.62 191.81 L 116.64 191.33 L 117.66 190.85 L 118.68 190.36 L 119.7 189.87 L 120.72 189.38 L 121.74 188.88 L 122.76 188.38 L 123.78 187.87 L 124.8 187.35 L 125.82 186.83 L 126.84 186.31 L 127.86 185.78 L 128.88 185.24 L 129.9 184.7 L 130.92 184.16 L 131.94 183.61 L 132.96 183.05 L 133.98 182.49 L 135 181.92 L 136.02 181.35 L 137.04 180.77 L 138.06 180.18 L 139.08 179.59 L 140.1 179 L 141.12 178.4 L 142.14 177.79 L 143.16 177.17 L 144.18 176.55 L 145.2 175.93 L 146.22 175.3 L 147.24 174.66 L 148.26 174.01 L 149.28 173.36 L 150.3 172.7 L 151.32 172.04 L 152.34 171.37 L 153.36 170.69 L 154.38 170.01 L 155.4 169.32 L 156.42 168.62 L 157.44 167.91 L 158.46 167.2 L 159.48 166.49 L 160.5 165.76 L 161.52 165.03 L 162.54 164.29 L 163.56 163.54 L 164.58 162.79 L 165.6 162.02 L 166.62 161.25 L 167.64 160.48 L 168.66 159.69 L 169.68 158.9 L 170.7 158.1 L 171.72 157.29 L 172.74 156.48 L 173.76 155.65 L 174.78 154.82 L 175.8 153.98 L 176.82 153.13 L 177.84 152.27 L 178.86 151.41 L 179.88 150.53 L 180.9 149.65 L 181.92 148.76 L 182.94 147.86 L 183.96 146.95 L 184.98 146.03 L 186 145.1 L 187.02 144.17 L 188.04 143.22 L 189.06 142.27 L 190.08 141.3 L 191.1 140.33 L 192.12 139.34 L 193.14 138.35 L 194.16 137.35 L 195.18 136.33 L 196.2 135.31 L 197.22 134.28 L 198.24 133.23 L 199.26 132.18 L 200.28 131.12 L 201.3 130.04 L 202.32 128.96 L 203.34 127.86 L 204.36 126.76 L 205.38 125.64 L 206.4 124.51 L 207.42 123.37 L 208.44 122.22 L 209.46 121.06 L 210.48 119.88 L 211.5 118.7 L 212.52 117.5 L 213.54 116.29 L 214.56 115.07 L 215.58 113.84 L 216.6 112.59 L 217.62 111.34 L 218.64 110.07 L 219.66 108.78 L 220.68 107.49 L 221.7 106.18 L 222.72 104.86 L 223.74 103.53 L 224.76 102.18 L 225.78 100.82 L 226.8 99.45 L 227.82 98.06 L 228.84 96.66 L 229.86 95.25 L 230.88 93.82 L 231.9 92.37 L 232.92 90.92 L 233.94 89.45 L 234.96 87.96 L 235.98 86.46 L 237 84.95 L 238.02 83.42 L 239.04 81.87 L 240.06 80.31 L 241.08 78.73 L 242.1 77.14 L 243.12 75.54 L 244.14 73.91 L 245.16 72.27 L 246.18 70.62 L 247.2 68.95 L 248.22 67.26 L 249.24 65.55 L 250.26 63.83 L 251.28 62.1 L 252.3 60.34 L 253.32 58.57 L 254.34 56.78 L 255.36 54.97 L 256.38 53.14 L 257.4 51.3 L 258.42 49.44 L 259.44 47.56 L 260.46 45.66 L 261.48 43.74 L 262.5 41.8 L 263.52 39.85 L 264.54 37.87 L 265.56 35.88 L 266.58 33.86 L 267.6 31.83 L 268.62 29.77 L 269.64 27.7 L 270.66 25.6 L 271.68 23.49 L 272.7 21.35 L 273.72 19.19 L 274.74 17.02 L 275.76 14.82 L 276.78 12.59 L 277.8 10.35 L 278.82 8.08 L 279.84 5.79 L 280.86 3.48 L 281.88 1.15 L 282.9 -1.21 L 283.92 -3.59 L 284.94 -5.99 L 285.96 -8.42 L 286.98 -10.87 L 288 -13.35\" class=\"fig-main fig-graph\" clip-path=\"url(#fig-doubling-time-clip)\"\u002F>\u003Ccircle cx=\"43.2\" cy=\"216\" r=\"3\" class=\"fig-dot\"\u002F>\u003Ctext x=\"51.2\" y=\"208\" text-anchor=\"start\" dominant-baseline=\"middle\" class=\"fig-label fig-upright\">N\u003Ctspan class=\"fig-sub\" dy=\"4\">0\u003C\u002Ftspan>\u003C\u002Ftext>\u003Ccircle cx=\"115.2\" cy=\"192\" r=\"3\" class=\"fig-dot\"\u002F>\u003Ccircle cx=\"187.2\" cy=\"144\" r=\"3\" class=\"fig-dot\"\u002F>\u003Ccircle cx=\"259.2\" cy=\"48\" r=\"3\" class=\"fig-dot\"\u002F>\u003Ctext x=\"115.2\" y=\"250.8\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">T\u003Ctspan class=\"fig-sub\" dy=\"4\">D\u003C\u002Ftspan>\u003C\u002Ftext>\u003Ctext x=\"187.2\" y=\"250.8\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">2T\u003Ctspan class=\"fig-sub\" dy=\"4\">D\u003C\u002Ftspan>\u003C\u002Ftext>\u003Ctext x=\"259.2\" y=\"250.8\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">3T\u003Ctspan class=\"fig-sub\" dy=\"4\">D\u003C\u002Ftspan>\u003C\u002Ftext>\u003C\u002Fsvg>","doubling-time","T_D = \\frac{\\ln 2}{\\ln q}","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>T\u003C\u002Fmi>\u003Cmi>D\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>=\u003C\u002Fmo>\u003Cmfrac>\u003Cmrow>\u003Cmi>ln\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cmrow>\u003Cmi>ln\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>q\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmfrac>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">T_D = \\frac{\\ln 2}{\\ln q}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3283em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0278em;\">D\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:2.2519em;vertical-align:-0.8804em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.3714em;\">\u003Cspan style=\"top:-2.314em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mop\">ln\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">q\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.677em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mop\">ln\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8804em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","middle",{},"\u002Fde\u002Fformulas\u002Fmath\u002Ffunctions\u002Fverdopplungszeit",[],[3698,3699,3700,3701],"exponential-growth","half-life","log-definition","exponential-function",{"title":548,"description":3585},"de\u002Fformulas\u002Fmath\u002Ffunctions\u002Fverdopplungszeit","Die Verdopplungszeit ist die feste Zeit, nach der sich ein exponentiell wachsender Bestand jeweils verdoppelt.",[3706,3707,3708,3709],"verdopplungszeit","exponentielles wachstum","wachstumsfaktor","logarithmus",[3711,3714,3717],{"symbol":608,"meaning":3712,"symbolHtml":3713},"Verdopplungszeit, gemessen in Zeitschritten","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>T\u003C\u002Fmi>\u003Cmi>D\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">T_D\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3283em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0278em;\">D\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":698,"meaning":3715,"symbolHtml":3716},"Wachstumsfaktor pro Zeitschritt, q = 1 + p\u002F100","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>q\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">q\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">q\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":1025,"meaning":3718,"symbolHtml":3719},"natürlicher Logarithmus (jede andere Basis liefert dasselbe Ergebnis)","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>ln\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\ln\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">ln\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","fOmistOLTrYgFXkBCKWrqWpFXmdPB1CHKJTcudjIZyQ",[3722,3725,3728,3731],{"stem":3723,"title":3724,"category":5},"de\u002Fformulas\u002Fmath\u002Falgebra\u002Flogarithmus-definition","Definition des Logarithmus",{"stem":3726,"title":3727,"category":208},"de\u002Fformulas\u002Fmath\u002Ffunctions\u002Fexponentialfunktion","Exponentialfunktion",{"stem":3729,"title":3730,"category":208},"de\u002Fformulas\u002Fmath\u002Ffunctions\u002Fexponentielles-wachstum","Exponentielles Wachstum",{"stem":3732,"title":3733,"category":208},"de\u002Fformulas\u002Fmath\u002Ffunctions\u002Fhalbwertszeit","Halbwertszeit",1793824275682]