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Eigenschaften:",[1171,1172,1173,1302,1565,1687],"ul",{},[1174,1175,1176,1239,1240,1301],"li",{},[574,1177,1179,1202],{"className":1178},[577],[574,1180,1182],{"className":1181},[581],[583,1183,1184],{"xmlns":585},[587,1185,1186,1199],{},[590,1187,1188,1190,1192,1195,1197],{},[593,1189,550],{},[596,1191,598],{},[655,1193,1194],{},"1",[596,1196,729],{},[655,1198,901],{},[603,1200,1201],{"encoding":605},"\\ln 1 = 0",[574,1203,1205,1229],{"className":1204,"ariaHidden":611},[610],[574,1206,1208,1211,1214,1217,1220,1223,1226],{"className":1207},[615],[574,1209],{"className":1210,"style":620},[619],[574,1212,550],{"className":1213},[624],[574,1215],{"className":1216,"style":629},[628],[574,1218,1194],{"className":1219},[633],[574,1221],{"className":1222,"style":677},[628],[574,1224,729],{"className":1225},[681],[574,1227],{"className":1228,"style":677},[628],[574,1230,1232,1236],{"className":1231},[615],[574,1233],{"className":1234,"style":1235},[619],"height:0.6444em;",[574,1237,901],{"className":1238},[633]," und ",[574,1241,1243,1265],{"className":1242},[577],[574,1244,1246],{"className":1245},[581],[583,1247,1248],{"xmlns":585},[587,1249,1250,1262],{},[590,1251,1252,1254,1256,1258,1260],{},[593,1253,550],{},[596,1255,598],{},[593,1257,650],{},[596,1259,729],{},[655,1261,1194],{},[603,1263,1264],{"encoding":605},"\\ln e = 1",[574,1266,1268,1292],{"className":1267,"ariaHidden":611},[610],[574,1269,1271,1274,1277,1280,1283,1286,1289],{"className":1270},[615],[574,1272],{"className":1273,"style":620},[619],[574,1275,550],{"className":1276},[624],[574,1278],{"className":1279,"style":629},[628],[574,1281,650],{"className":1282},[633,634],[574,1284],{"className":1285,"style":677},[628],[574,1287,729],{"className":1288},[681],[574,1290],{"className":1291,"style":677},[628],[574,1293,1295,1298],{"className":1294},[615],[574,1296],{"className":1297,"style":1235},[619],[574,1299,1194],{"className":1300},[633],".",[1174,1303,1304,1305,1381,1382,1420,1421,1472,1473,1381,1526,1564],{},"Für ",[574,1306,1308,1331],{"className":1307},[577],[574,1309,1311],{"className":1310},[581],[583,1312,1313],{"xmlns":585},[587,1314,1315,1328],{},[590,1316,1317,1319,1322,1324,1326],{},[655,1318,901],{},[596,1320,1321],{},"\u003C",[593,1323,601],{},[596,1325,1321],{},[655,1327,1194],{},[603,1329,1330],{"encoding":605},"0 \u003C x \u003C 1",[574,1332,1334,1353,1372],{"className":1333,"ariaHidden":611},[610],[574,1335,1337,1341,1344,1347,1350],{"className":1336},[615],[574,1338],{"className":1339,"style":1340},[619],"height:0.6835em;vertical-align:-0.0391em;",[574,1342,901],{"className":1343},[633],[574,1345],{"className":1346,"style":677},[628],[574,1348,1321],{"className":1349},[681],[574,1351],{"className":1352,"style":677},[628],[574,1354,1356,1360,1363,1366,1369],{"className":1355},[615],[574,1357],{"className":1358,"style":1359},[619],"height:0.5782em;vertical-align:-0.0391em;",[574,1361,601],{"className":1362},[633,634],[574,1364],{"className":1365,"style":677},[628],[574,1367,1321],{"className":1368},[681],[574,1370],{"className":1371,"style":677},[628],[574,1373,1375,1378],{"className":1374},[615],[574,1376],{"className":1377,"style":1235},[619],[574,1379,1194],{"className":1380},[633]," ist ",[574,1383,1385,1402],{"className":1384},[577],[574,1386,1388],{"className":1387},[581],[583,1389,1390],{"xmlns":585},[587,1391,1392,1400],{},[590,1393,1394,1396,1398],{},[593,1395,550],{},[596,1397,598],{},[593,1399,601],{},[603,1401,606],{"encoding":605},[574,1403,1405],{"className":1404,"ariaHidden":611},[610],[574,1406,1408,1411,1414,1417],{"className":1407},[615],[574,1409],{"className":1410,"style":620},[619],[574,1412,550],{"className":1413},[624],[574,1415],{"className":1416,"style":629},[628],[574,1418,601],{"className":1419},[633,634]," negativ, für ",[574,1422,1424,1442],{"className":1423},[577],[574,1425,1427],{"className":1426},[581],[583,1428,1429],{"xmlns":585},[587,1430,1431,1439],{},[590,1432,1433,1435,1437],{},[593,1434,601],{},[596,1436,898],{},[655,1438,1194],{},[603,1440,1441],{"encoding":605},"x > 1",[574,1443,1445,1463],{"className":1444,"ariaHidden":611},[610],[574,1446,1448,1451,1454,1457,1460],{"className":1447},[615],[574,1449],{"className":1450,"style":1359},[619],[574,1452,601],{"className":1453},[633,634],[574,1455],{"className":1456,"style":677},[628],[574,1458,898],{"className":1459},[681],[574,1461],{"className":1462,"style":677},[628],[574,1464,1466,1469],{"className":1465},[615],[574,1467],{"className":1468,"style":1235},[619],[574,1470,1194],{"className":1471},[633]," positiv. Für ",[574,1474,1476,1495],{"className":1475},[577],[574,1477,1479],{"className":1478},[581],[583,1480,1481],{"xmlns":585},[587,1482,1483,1492],{},[590,1484,1485,1487,1490],{},[593,1486,601],{},[596,1488,1489],{},"≤",[655,1491,901],{},[603,1493,1494],{"encoding":605},"x \\leq 0",[574,1496,1498,1517],{"className":1497,"ariaHidden":611},[610],[574,1499,1501,1505,1508,1511,1514],{"className":1500},[615],[574,1502],{"className":1503,"style":1504},[619],"height:0.7719em;vertical-align:-0.136em;",[574,1506,601],{"className":1507},[633,634],[574,1509],{"className":1510,"style":677},[628],[574,1512,1489],{"className":1513},[681],[574,1515],{"className":1516,"style":677},[628],[574,1518,1520,1523],{"className":1519},[615],[574,1521],{"className":1522,"style":1235},[619],[574,1524,901],{"className":1525},[633],[574,1527,1529,1546],{"className":1528},[577],[574,1530,1532],{"className":1531},[581],[583,1533,1534],{"xmlns":585},[587,1535,1536,1544],{},[590,1537,1538,1540,1542],{},[593,1539,550],{},[596,1541,598],{},[593,1543,601],{},[603,1545,606],{"encoding":605},[574,1547,1549],{"className":1548,"ariaHidden":611},[610],[574,1550,1552,1555,1558,1561],{"className":1551},[615],[574,1553],{"className":1554,"style":620},[619],[574,1556,550],{"className":1557},[624],[574,1559],{"className":1560,"style":629},[628],[574,1562,601],{"className":1563},[633,634]," nicht definiert.",[1174,1566,1567,1568,1301],{},"Die Logarithmusgesetze gelten wie bei jeder anderen Basis, z.B. ",[574,1569,1571,1614],{"className":1570},[577],[574,1572,1574],{"className":1573},[581],[583,1575,1576],{"xmlns":585},[587,1577,1578,1611],{},[590,1579,1580,1582,1584,1586,1589,1592,1594,1596,1598,1600,1602,1605,1607,1609],{},[593,1581,550],{},[596,1583,598],{},[596,1585,893],{"stretchy":892},[593,1587,1588],{},"a",[593,1590,1591],{},"b",[596,1593,904],{"stretchy":892},[596,1595,729],{},[593,1597,550],{},[596,1599,598],{},[593,1601,1588],{},[596,1603,1604],{},"+",[593,1606,550],{},[596,1608,598],{},[593,1610,1591],{},[603,1612,1613],{"encoding":605},"\\ln(ab) = \\ln a + \\ln b",[574,1615,1617,1645,1672],{"className":1616,"ariaHidden":611},[610],[574,1618,1620,1623,1626,1629,1633,1636,1639,1642],{"className":1619},[615],[574,1621],{"className":1622,"style":1022},[619],[574,1624,550],{"className":1625},[624],[574,1627,893],{"className":1628},[1033],[574,1630,1632],{"className":1631},[633,634],"ab",[574,1634,904],{"className":1635},[1058],[574,1637],{"className":1638,"style":677},[628],[574,1640,729],{"className":1641},[681],[574,1643],{"className":1644,"style":677},[628],[574,1646,1648,1652,1655,1658,1661,1665,1669],{"className":1647},[615],[574,1649],{"className":1650,"style":1651},[619],"height:0.7778em;vertical-align:-0.0833em;",[574,1653,550],{"className":1654},[624],[574,1656],{"className":1657,"style":629},[628],[574,1659,1588],{"className":1660},[633,634],[574,1662],{"className":1663,"style":1664},[628],"margin-right:0.2222em;",[574,1666,1604],{"className":1667},[1668],"mbin",[574,1670],{"className":1671,"style":1664},[628],[574,1673,1675,1678,1681,1684],{"className":1674},[615],[574,1676],{"className":1677,"style":620},[619],[574,1679,550],{"className":1680},[624],[574,1682],{"className":1683,"style":629},[628],[574,1685,1591],{"className":1686},[633,634],[1174,1688,1689,1690,1721,1722,1301],{},"Jeden anderen Logarithmus kann man mit ",[574,1691,1693,1709],{"className":1692},[577],[574,1694,1696],{"className":1695},[581],[583,1697,1698],{"xmlns":585},[587,1699,1700,1706],{},[590,1701,1702,1704],{},[593,1703,550],{},[596,1705,598],{},[603,1707,1708],{"encoding":605},"\\ln",[574,1710,1712],{"className":1711,"ariaHidden":611},[610],[574,1713,1715,1718],{"className":1714},[615],[574,1716],{"className":1717,"style":620},[619],[574,1719,550],{"className":1720},[624]," ausdrücken: ",[574,1723,1725,1773],{"className":1724},[577],[574,1726,1728],{"className":1727},[581],[583,1729,1730],{"xmlns":585},[587,1731,1732,1770],{},[590,1733,1734,1744,1746,1748],{},[731,1735,1736,1742],{},[590,1737,1738,1740],{},[593,1739,737],{},[596,1741,598],{},[593,1743,1588],{},[593,1745,1591],{},[596,1747,729],{},[1749,1750,1751],"mstyle",{"scriptlevel":901,"displaystyle":611},[1752,1753,1754,1762],"mfrac",{},[590,1755,1756,1758,1760],{},[593,1757,550],{},[596,1759,598],{},[593,1761,1591],{},[590,1763,1764,1766,1768],{},[593,1765,550],{},[596,1767,598],{},[593,1769,1588],{},[603,1771,1772],{"encoding":605},"\\log_a b = \\dfrac{\\ln b}{\\ln a}",[574,1774,1776,1839],{"className":1775,"ariaHidden":611},[610],[574,1777,1779,1782,1824,1827,1830,1833,1836],{"className":1778},[615],[574,1780],{"className":1781,"style":780},[619],[574,1783,1785,1790],{"className":1784},[624],[574,1786,787,1788],{"className":1787},[624],[574,1789,791],{"style":790},[574,1791,1793],{"className":1792},[795],[574,1794,1796,1816],{"className":1795},[799,800],[574,1797,1799,1813],{"className":1798},[804],[574,1800,1802],{"className":1801,"style":809},[808],[574,1803,1804,1807],{"style":812},[574,1805],{"className":1806,"style":817},[816],[574,1808,1810],{"className":1809},[821,822,823,824],[574,1811,1588],{"className":1812},[633,634,824],[574,1814,832],{"className":1815},[831],[574,1817,1819],{"className":1818},[804],[574,1820,1822],{"className":1821,"style":839},[808],[574,1823],{},[574,1825],{"className":1826,"style":629},[628],[574,1828,1591],{"className":1829},[633,634],[574,1831],{"className":1832,"style":677},[628],[574,1834,729],{"className":1835},[681],[574,1837],{"className":1838,"style":677},[628],[574,1840,1842,1846],{"className":1841},[615],[574,1843],{"className":1844,"style":1845},[619],"height:2.0574em;vertical-align:-0.686em;",[574,1847,1849,1853,1926],{"className":1848},[633],[574,1850],{"className":1851},[1033,1852],"nulldelimiter",[574,1854,1856],{"className":1855},[1752],[574,1857,1859,1917],{"className":1858},[799,800],[574,1860,1862,1914],{"className":1861},[804],[574,1863,1866,1885,1896],{"className":1864,"style":1865},[808],"height:1.3714em;",[574,1867,1869,1873],{"style":1868},"top:-2.314em;",[574,1870],{"className":1871,"style":1872},[816],"height:3em;",[574,1874,1876,1879,1882],{"className":1875},[633],[574,1877,550],{"className":1878},[624],[574,1880],{"className":1881,"style":629},[628],[574,1883,1588],{"className":1884},[633,634],[574,1886,1888,1891],{"style":1887},"top:-3.23em;",[574,1889],{"className":1890,"style":1872},[816],[574,1892],{"className":1893,"style":1895},[1894],"frac-line","border-bottom-width:0.04em;",[574,1897,1899,1902],{"style":1898},"top:-3.677em;",[574,1900],{"className":1901,"style":1872},[816],[574,1903,1905,1908,1911],{"className":1904},[633],[574,1906,550],{"className":1907},[624],[574,1909],{"className":1910,"style":629},[628],[574,1912,1591],{"className":1913},[633,634],[574,1915,832],{"className":1916},[831],[574,1918,1920],{"className":1919},[804],[574,1921,1924],{"className":1922,"style":1923},[808],"height:0.686em;",[574,1925],{},[574,1927],{"className":1928},[1058,1852],[565,1930,1931,1932,1962,1963,1239,2024,2062],{},"In der Analysis ist ",[574,1933,1935,1950],{"className":1934},[577],[574,1936,1938],{"className":1937},[581],[583,1939,1940],{"xmlns":585},[587,1941,1942,1948],{},[590,1943,1944,1946],{},[593,1945,550],{},[596,1947,598],{},[603,1949,1708],{"encoding":605},[574,1951,1953],{"className":1952,"ariaHidden":611},[610],[574,1954,1956,1959],{"className":1955},[615],[574,1957],{"className":1958,"style":620},[619],[574,1960,550],{"className":1961},[624]," der Standardlogarithmus, weil ",[574,1964,1966,1984],{"className":1965},[577],[574,1967,1969],{"className":1968},[581],[583,1970,1971],{"xmlns":585},[587,1972,1973,1981],{},[590,1974,1975],{},[871,1976,1977,1979],{},[593,1978,650],{},[593,1980,601],{},[603,1982,1983],{"encoding":605},"e^x",[574,1985,1987],{"className":1986,"ariaHidden":611},[610],[574,1988,1990,1994],{"className":1989},[615],[574,1991],{"className":1992,"style":1993},[619],"height:0.6644em;",[574,1995,1997,2000],{"className":1996},[633],[574,1998,650],{"className":1999},[633,634],[574,2001,2003],{"className":2002},[795],[574,2004,2006],{"className":2005},[799],[574,2007,2009],{"className":2008},[804],[574,2010,2012],{"className":2011,"style":1993},[808],[574,2013,2015,2018],{"style":2014},"top:-3.063em;margin-right:0.05em;",[574,2016],{"className":2017,"style":817},[816],[574,2019,2021],{"className":2020},[821,822,823,824],[574,2022,601],{"className":2023},[633,634,824],[574,2025,2027,2044],{"className":2026},[577],[574,2028,2030],{"className":2029},[581],[583,2031,2032],{"xmlns":585},[587,2033,2034,2042],{},[590,2035,2036,2038,2040],{},[593,2037,550],{},[596,2039,598],{},[593,2041,601],{},[603,2043,606],{"encoding":605},[574,2045,2047],{"className":2046,"ariaHidden":611},[610],[574,2048,2050,2053,2056,2059],{"className":2049},[615],[574,2051],{"className":2052,"style":620},[619],[574,2054,550],{"className":2055},[624],[574,2057],{"className":2058,"style":629},[628],[574,2060,601],{"className":2061},[633,634]," besonders einfache Ableitungen haben.",[560,2064,2066],{"id":2065},"beispiele","Beispiele",[574,2068,2070],{"className":2069},[856],[574,2071,2073,2141],{"className":2072},[577],[574,2074,2076],{"className":2075},[581],[583,2077,2078],{"xmlns":585,"display":865},[587,2079,2080,2138],{},[590,2081,2082,2084,2086,2099,2101,2103,2105,2107,2109,2111,2114,2116,2119,2121,2123,2125,2127,2130,2132,2135],{},[593,2083,550],{},[596,2085,598],{},[590,2087,2088,2090,2097],{},[596,2089,893],{"fence":611},[871,2091,2092,2094],{},[593,2093,650],{},[655,2095,2096],{},"3",[596,2098,904],{"fence":611},[596,2100,729],{},[655,2102,2096],{},[596,2104,703],{"separator":611},[628,2106],{"width":909},[593,2108,550],{},[596,2110,598],{},[655,2112,2113],{},"10",[596,2115,653],{},[655,2117,2118],{},"2,3026",[596,2120,703],{"separator":611},[628,2122],{"width":909},[593,2124,550],{},[596,2126,598],{},[655,2128,2129],{},"0,5",[596,2131,653],{},[596,2133,2134],{},"−",[655,2136,2137],{},"0,6931",[603,2139,2140],{"encoding":605},"\\ln\\left(e^3\\right) = 3, \\qquad \\ln 10 \\approx 2{,}3026, \\qquad \\ln 0{,}5 \\approx -0{,}6931",[574,2142,2144,2213,2250,2306],{"className":2143,"ariaHidden":611},[610],[574,2145,2147,2151,2154,2157,2204,2207,2210],{"className":2146},[615],[574,2148],{"className":2149,"style":2150},[619],"height:1.2141em;vertical-align:-0.35em;",[574,2152,550],{"className":2153},[624],[574,2155],{"className":2156,"style":629},[628],[574,2158,2160,2168,2198],{"className":2159},[1078],[574,2161,2163],{"className":2162,"style":1083},[1033,1082],[574,2164,893],{"className":2165},[2166,2167],"delimsizing","size1",[574,2169,2171,2174],{"className":2170},[633],[574,2172,650],{"className":2173},[633,634],[574,2175,2177],{"className":2176},[795],[574,2178,2180],{"className":2179},[799],[574,2181,2183],{"className":2182},[804],[574,2184,2187],{"className":2185,"style":2186},[808],"height:0.8641em;",[574,2188,2189,2192],{"style":979},[574,2190],{"className":2191,"style":817},[816],[574,2193,2195],{"className":2194},[821,822,823,824],[574,2196,2096],{"className":2197},[633,824],[574,2199,2201],{"className":2200,"style":1083},[1058,1082],[574,2202,904],{"className":2203},[2166,2167],[574,2205],{"className":2206,"style":677},[628],[574,2208,729],{"className":2209},[681],[574,2211],{"className":2212,"style":677},[628],[574,2214,2216,2220,2223,2226,2229,2232,2235,2238,2241,2244,2247],{"className":2215},[615],[574,2217],{"className":2218,"style":2219},[619],"height:0.8889em;vertical-align:-0.1944em;",[574,2221,2096],{"className":2222},[633],[574,2224,703],{"className":2225},[702],[574,2227],{"className":2228,"style":1065},[628],[574,2230],{"className":2231,"style":629},[628],[574,2233,550],{"className":2234},[624],[574,2236],{"className":2237,"style":629},[628],[574,2239,2113],{"className":2240},[633],[574,2242],{"className":2243,"style":677},[628],[574,2245,653],{"className":2246},[681],[574,2248],{"className":2249,"style":677},[628],[574,2251,2253,2256,2259,2265,2269,2272,2275,2278,2281,2284,2287,2293,2297,2300,2303],{"className":2252},[615],[574,2254],{"className":2255,"style":2219},[619],[574,2257,695],{"className":2258},[633],[574,2260,2262],{"className":2261},[633],[574,2263,703],{"className":2264},[702],[574,2266,2268],{"className":2267},[633],"3026",[574,2270,703],{"className":2271},[702],[574,2273],{"className":2274,"style":1065},[628],[574,2276],{"className":2277,"style":629},[628],[574,2279,550],{"className":2280},[624],[574,2282],{"className":2283,"style":629},[628],[574,2285,901],{"className":2286},[633],[574,2288,2290],{"className":2289},[633],[574,2291,703],{"className":2292},[702],[574,2294,2296],{"className":2295},[633],"5",[574,2298],{"className":2299,"style":677},[628],[574,2301,653],{"className":2302},[681],[574,2304],{"className":2305,"style":677},[628],[574,2307,2309,2312,2315,2318,2324],{"className":2308},[615],[574,2310],{"className":2311,"style":691},[619],[574,2313,2134],{"className":2314},[633],[574,2316,901],{"className":2317},[633],[574,2319,2321],{"className":2320},[633],[574,2322,703],{"className":2323},[702],[574,2325,2327],{"className":2326},[633],"6931",[565,2329,2330,2331,2424],{},"Gleichung mit der e-Funktion: ",[574,2332,2334,2361],{"className":2333},[577],[574,2335,2337],{"className":2336},[581],[583,2338,2339],{"xmlns":585},[587,2340,2341,2358],{},[590,2342,2343,2353,2355],{},[871,2344,2345,2347],{},[593,2346,650],{},[590,2348,2349,2351],{},[655,2350,695],{},[593,2352,601],{},[596,2354,729],{},[655,2356,2357],{},"7",[603,2359,2360],{"encoding":605},"e^{2x} = 7",[574,2362,2364,2415],{"className":2363,"ariaHidden":611},[610],[574,2365,2367,2371,2406,2409,2412],{"className":2366},[615],[574,2368],{"className":2369,"style":2370},[619],"height:0.8141em;",[574,2372,2374,2377],{"className":2373},[633],[574,2375,650],{"className":2376},[633,634],[574,2378,2380],{"className":2379},[795],[574,2381,2383],{"className":2382},[799],[574,2384,2386],{"className":2385},[804],[574,2387,2389],{"className":2388,"style":2370},[808],[574,2390,2391,2394],{"style":2014},[574,2392],{"className":2393,"style":817},[816],[574,2395,2397],{"className":2396},[821,822,823,824],[574,2398,2400,2403],{"className":2399},[633,824],[574,2401,695],{"className":2402},[633,824],[574,2404,601],{"className":2405},[633,634,824],[574,2407],{"className":2408,"style":677},[628],[574,2410,729],{"className":2411},[681],[574,2413],{"className":2414,"style":677},[628],[574,2416,2418,2421],{"className":2417},[615],[574,2419],{"className":2420,"style":1235},[619],[574,2422,2357],{"className":2423},[633],". Man logarithmiert beide Seiten:",[574,2426,2428],{"className":2427},[856],[574,2429,2431,2483],{"className":2430},[577],[574,2432,2434],{"className":2433},[581],[583,2435,2436],{"xmlns":585,"display":865},[587,2437,2438,2480],{},[590,2439,2440,2442,2444,2446,2448,2450,2452,2454,2457,2459,2461,2463,2475,2477],{},[655,2441,695],{},[593,2443,601],{},[596,2445,729],{},[593,2447,550],{},[596,2449,598],{},[655,2451,2357],{},[628,2453],{"width":889},[596,2455,2456],{},"⇒",[628,2458],{"width":889},[593,2460,601],{},[596,2462,729],{},[1752,2464,2465,2473],{},[590,2466,2467,2469,2471],{},[593,2468,550],{},[596,2470,598],{},[655,2472,2357],{},[655,2474,695],{},[596,2476,653],{},[655,2478,2479],{},"0,9730",[603,2481,2482],{"encoding":605},"2x = \\ln 7 \\quad\\Rightarrow\\quad x = \\frac{\\ln 7}{2} \\approx 0{,}9730",[574,2484,2486,2507,2537,2556,2639],{"className":2485,"ariaHidden":611},[610],[574,2487,2489,2492,2495,2498,2501,2504],{"className":2488},[615],[574,2490],{"className":2491,"style":1235},[619],[574,2493,695],{"className":2494},[633],[574,2496,601],{"className":2497},[633,634],[574,2499],{"className":2500,"style":677},[628],[574,2502,729],{"className":2503},[681],[574,2505],{"className":2506,"style":677},[628],[574,2508,2510,2513,2516,2519,2522,2525,2528,2531,2534],{"className":2509},[615],[574,2511],{"className":2512,"style":620},[619],[574,2514,550],{"className":2515},[624],[574,2517],{"className":2518,"style":629},[628],[574,2520,2357],{"className":2521},[633],[574,2523],{"className":2524,"style":1029},[628],[574,2526],{"className":2527,"style":677},[628],[574,2529,2456],{"className":2530},[681],[574,2532],{"className":2533,"style":1029},[628],[574,2535],{"className":2536,"style":677},[628],[574,2538,2540,2544,2547,2550,2553],{"className":2539},[615],[574,2541],{"className":2542,"style":2543},[619],"height:0.4306em;",[574,2545,601],{"className":2546},[633,634],[574,2548],{"className":2549,"style":677},[628],[574,2551,729],{"className":2552},[681],[574,2554],{"className":2555,"style":677},[628],[574,2557,2559,2562,2630,2633,2636],{"className":2558},[615],[574,2560],{"className":2561,"style":1845},[619],[574,2563,2565,2568,2627],{"className":2564},[633],[574,2566],{"className":2567},[1033,1852],[574,2569,2571],{"className":2570},[1752],[574,2572,2574,2619],{"className":2573},[799,800],[574,2575,2577,2616],{"className":2576},[804],[574,2578,2580,2591,2599],{"className":2579,"style":1865},[808],[574,2581,2582,2585],{"style":1868},[574,2583],{"className":2584,"style":1872},[816],[574,2586,2588],{"className":2587},[633],[574,2589,695],{"className":2590},[633],[574,2592,2593,2596],{"style":1887},[574,2594],{"className":2595,"style":1872},[816],[574,2597],{"className":2598,"style":1895},[1894],[574,2600,2601,2604],{"style":1898},[574,2602],{"className":2603,"style":1872},[816],[574,2605,2607,2610,2613],{"className":2606},[633],[574,2608,550],{"className":2609},[624],[574,2611],{"className":2612,"style":629},[628],[574,2614,2357],{"className":2615},[633],[574,2617,832],{"className":2618},[831],[574,2620,2622],{"className":2621},[804],[574,2623,2625],{"className":2624,"style":1923},[808],[574,2626],{},[574,2628],{"className":2629},[1058,1852],[574,2631],{"className":2632,"style":677},[628],[574,2634,653],{"className":2635},[681],[574,2637],{"className":2638,"style":677},[628],[574,2640,2642,2645,2648,2654],{"className":2641},[615],[574,2643],{"className":2644,"style":691},[619],[574,2646,901],{"className":2647},[633],[574,2649,2651],{"className":2650},[633],[574,2652,703],{"className":2653},[702],[574,2655,2657],{"className":2656},[633],"9730",[565,2659,2660],{},"Logarithmus zu einer anderen Basis berechnen:",[574,2662,2664],{"className":2663},[856],[574,2665,2667,2716],{"className":2666},[577],[574,2668,2670],{"className":2669},[581],[583,2671,2672],{"xmlns":585,"display":865},[587,2673,2674,2713],{},[590,2675,2676,2686,2688,2690,2708,2710],{},[731,2677,2678,2684],{},[590,2679,2680,2682],{},[593,2681,737],{},[596,2683,598],{},[655,2685,695],{},[655,2687,2113],{},[596,2689,729],{},[1752,2691,2692,2700],{},[590,2693,2694,2696,2698],{},[593,2695,550],{},[596,2697,598],{},[655,2699,2113],{},[590,2701,2702,2704,2706],{},[593,2703,550],{},[596,2705,598],{},[655,2707,695],{},[596,2709,653],{},[655,2711,2712],{},"3,3219",[603,2714,2715],{"encoding":605},"\\log_2 10 = \\frac{\\ln 10}{\\ln 2} \\approx 3{,}3219",[574,2717,2719,2783,2872],{"className":2718,"ariaHidden":611},[610],[574,2720,2722,2725,2768,2771,2774,2777,2780],{"className":2721},[615],[574,2723],{"className":2724,"style":780},[619],[574,2726,2728,2733],{"className":2727},[624],[574,2729,787,2731],{"className":2730},[624],[574,2732,791],{"style":790},[574,2734,2736],{"className":2735},[795],[574,2737,2739,2760],{"className":2738},[799,800],[574,2740,2742,2757],{"className":2741},[804],[574,2743,2746],{"className":2744,"style":2745},[808],"height:0.207em;",[574,2747,2748,2751],{"style":812},[574,2749],{"className":2750,"style":817},[816],[574,2752,2754],{"className":2753},[821,822,823,824],[574,2755,695],{"className":2756},[633,824],[574,2758,832],{"className":2759},[831],[574,2761,2763],{"className":2762},[804],[574,2764,2766],{"className":2765,"style":839},[808],[574,2767],{},[574,2769],{"className":2770,"style":629},[628],[574,2772,2113],{"className":2773},[633],[574,2775],{"className":2776,"style":677},[628],[574,2778,729],{"className":2779},[681],[574,2781],{"className":2782,"style":677},[628],[574,2784,2786,2789,2863,2866,2869],{"className":2785},[615],[574,2787],{"className":2788,"style":1845},[619],[574,2790,2792,2795,2860],{"className":2791},[633],[574,2793],{"className":2794},[1033,1852],[574,2796,2798],{"className":2797},[1752],[574,2799,2801,2852],{"className":2800},[799,800],[574,2802,2804,2849],{"className":2803},[804],[574,2805,2807,2824,2832],{"className":2806,"style":1865},[808],[574,2808,2809,2812],{"style":1868},[574,2810],{"className":2811,"style":1872},[816],[574,2813,2815,2818,2821],{"className":2814},[633],[574,2816,550],{"className":2817},[624],[574,2819],{"className":2820,"style":629},[628],[574,2822,695],{"className":2823},[633],[574,2825,2826,2829],{"style":1887},[574,2827],{"className":2828,"style":1872},[816],[574,2830],{"className":2831,"style":1895},[1894],[574,2833,2834,2837],{"style":1898},[574,2835],{"className":2836,"style":1872},[816],[574,2838,2840,2843,2846],{"className":2839},[633],[574,2841,550],{"className":2842},[624],[574,2844],{"className":2845,"style":629},[628],[574,2847,2113],{"className":2848},[633],[574,2850,832],{"className":2851},[831],[574,2853,2855],{"className":2854},[804],[574,2856,2858],{"className":2857,"style":1923},[808],[574,2859],{},[574,2861],{"className":2862},[1058,1852],[574,2864],{"className":2865,"style":677},[628],[574,2867,653],{"className":2868},[681],[574,2870],{"className":2871,"style":677},[628],[574,2873,2875,2878,2881,2887],{"className":2874},[615],[574,2876],{"className":2877,"style":691},[619],[574,2879,2096],{"className":2880},[633],[574,2882,2884],{"className":2883},[633],[574,2885,703],{"className":2886},[702],[574,2888,2890],{"className":2889},[633],"3219",[560,2892,2894],{"id":2893},"typische-fehler","Typische Fehler",[1171,2896,2897,3179,3252,3350],{},[1174,2898,2899,2900,3074,3075,1301],{},"Den Logarithmus einer Summe aufteilen: ",[574,2901,2903,2947],{"className":2902},[577],[574,2904,2906],{"className":2905},[581],[583,2907,2908],{"xmlns":585},[587,2909,2910,2944],{},[590,2911,2912,2914,2916,2918,2920,2922,2924,2926,2930,2932,2934,2936,2938,2940,2942],{},[593,2913,550],{},[596,2915,598],{},[596,2917,893],{"stretchy":892},[655,2919,695],{},[596,2921,1604],{},[655,2923,2096],{},[596,2925,904],{"stretchy":892},[596,2927,2929],{"mathvariant":2928},"normal","≠",[593,2931,550],{},[596,2933,598],{},[655,2935,695],{},[596,2937,1604],{},[593,2939,550],{},[596,2941,598],{},[655,2943,2096],{},[603,2945,2946],{"encoding":605},"\\ln(2 + 3) \\neq \\ln 2 + \\ln 3",[574,2948,2950,2974,3035,3059],{"className":2949,"ariaHidden":611},[610],[574,2951,2953,2956,2959,2962,2965,2968,2971],{"className":2952},[615],[574,2954],{"className":2955,"style":1022},[619],[574,2957,550],{"className":2958},[624],[574,2960,893],{"className":2961},[1033],[574,2963,695],{"className":2964},[633],[574,2966],{"className":2967,"style":1664},[628],[574,2969,1604],{"className":2970},[1668],[574,2972],{"className":2973,"style":1664},[628],[574,2975,2977,2980,2983,2986,2989,3032],{"className":2976},[615],[574,2978],{"className":2979,"style":1022},[619],[574,2981,2096],{"className":2982},[633],[574,2984,904],{"className":2985},[1058],[574,2987],{"className":2988,"style":677},[628],[574,2990,2992,3025,3029],{"className":2991},[681],[574,2993,2995],{"className":2994},[681],[574,2996,2999],{"className":2997},[633,2998],"vbox",[574,3000,3003],{"className":3001},[3002],"thinbox",[574,3004,3007,3010,3021],{"className":3005},[3006],"rlap",[574,3008],{"className":3009,"style":2219},[619],[574,3011,3014],{"className":3012},[3013],"inner",[574,3015,3017],{"className":3016},[633],[574,3018,3020],{"className":3019},[681],"",[574,3022],{"className":3023},[3024],"fix",[574,3026],{"className":3027},[628,3028],"nobreak",[574,3030,729],{"className":3031},[681],[574,3033],{"className":3034,"style":677},[628],[574,3036,3038,3041,3044,3047,3050,3053,3056],{"className":3037},[615],[574,3039],{"className":3040,"style":1651},[619],[574,3042,550],{"className":3043},[624],[574,3045],{"className":3046,"style":629},[628],[574,3048,695],{"className":3049},[633],[574,3051],{"className":3052,"style":1664},[628],[574,3054,1604],{"className":3055},[1668],[574,3057],{"className":3058,"style":1664},[628],[574,3060,3062,3065,3068,3071],{"className":3061},[615],[574,3063],{"className":3064,"style":620},[619],[574,3066,550],{"className":3067},[624],[574,3069],{"className":3070,"style":629},[628],[574,3072,2096],{"className":3073},[633],". Richtig ist ",[574,3076,3078,3113],{"className":3077},[577],[574,3079,3081],{"className":3080},[581],[583,3082,3083],{"xmlns":585},[587,3084,3085,3110],{},[590,3086,3087,3089,3091,3093,3095,3097,3099,3101,3103,3105,3107],{},[593,3088,550],{},[596,3090,598],{},[655,3092,695],{},[596,3094,1604],{},[593,3096,550],{},[596,3098,598],{},[655,3100,2096],{},[596,3102,729],{},[593,3104,550],{},[596,3106,598],{},[655,3108,3109],{},"6",[603,3111,3112],{"encoding":605},"\\ln 2 + \\ln 3 = \\ln 6",[574,3114,3116,3140,3164],{"className":3115,"ariaHidden":611},[610],[574,3117,3119,3122,3125,3128,3131,3134,3137],{"className":3118},[615],[574,3120],{"className":3121,"style":1651},[619],[574,3123,550],{"className":3124},[624],[574,3126],{"className":3127,"style":629},[628],[574,3129,695],{"className":3130},[633],[574,3132],{"className":3133,"style":1664},[628],[574,3135,1604],{"className":3136},[1668],[574,3138],{"className":3139,"style":1664},[628],[574,3141,3143,3146,3149,3152,3155,3158,3161],{"className":3142},[615],[574,3144],{"className":3145,"style":620},[619],[574,3147,550],{"className":3148},[624],[574,3150],{"className":3151,"style":629},[628],[574,3153,2096],{"className":3154},[633],[574,3156],{"className":3157,"style":677},[628],[574,3159,729],{"className":3160},[681],[574,3162],{"className":3163,"style":677},[628],[574,3165,3167,3170,3173,3176],{"className":3166},[615],[574,3168],{"className":3169,"style":620},[619],[574,3171,550],{"className":3172},[624],[574,3174],{"className":3175,"style":629},[628],[574,3177,3109],{"className":3178},[633],[1174,3180,3181,3220,3221,3251],{},[574,3182,3184,3202],{"className":3183},[577],[574,3185,3187],{"className":3186},[581],[583,3188,3189],{"xmlns":585},[587,3190,3191,3199],{},[590,3192,3193,3195,3197],{},[593,3194,550],{},[596,3196,598],{},[655,3198,901],{},[603,3200,3201],{"encoding":605},"\\ln 0",[574,3203,3205],{"className":3204,"ariaHidden":611},[610],[574,3206,3208,3211,3214,3217],{"className":3207},[615],[574,3209],{"className":3210,"style":620},[619],[574,3212,550],{"className":3213},[624],[574,3215],{"className":3216,"style":629},[628],[574,3218,901],{"className":3219},[633]," oder ",[574,3222,3224,3239],{"className":3223},[577],[574,3225,3227],{"className":3226},[581],[583,3228,3229],{"xmlns":585},[587,3230,3231,3237],{},[590,3232,3233,3235],{},[593,3234,550],{},[596,3236,598],{},[603,3238,1708],{"encoding":605},[574,3240,3242],{"className":3241,"ariaHidden":611},[610],[574,3243,3245,3248],{"className":3244},[615],[574,3246],{"className":3247,"style":620},[619],[574,3249,550],{"className":3250},[624]," einer negativen Zahl berechnen wollen. Beides ist in den reellen Zahlen nicht definiert.",[1174,3253,3254,3255,3345,3346,3349],{},"Beim Lösen von ",[574,3256,3258,3283],{"className":3257},[577],[574,3259,3261],{"className":3260},[581],[583,3262,3263],{"xmlns":585},[587,3264,3265,3281],{},[590,3266,3267,3277,3279],{},[871,3268,3269,3271],{},[593,3270,650],{},[590,3272,3273,3275],{},[655,3274,695],{},[593,3276,601],{},[596,3278,729],{},[655,3280,2357],{},[603,3282,2360],{"encoding":605},[574,3284,3286,3336],{"className":3285,"ariaHidden":611},[610],[574,3287,3289,3292,3327,3330,3333],{"className":3288},[615],[574,3290],{"className":3291,"style":2370},[619],[574,3293,3295,3298],{"className":3294},[633],[574,3296,650],{"className":3297},[633,634],[574,3299,3301],{"className":3300},[795],[574,3302,3304],{"className":3303},[799],[574,3305,3307],{"className":3306},[804],[574,3308,3310],{"className":3309,"style":2370},[808],[574,3311,3312,3315],{"style":2014},[574,3313],{"className":3314,"style":817},[816],[574,3316,3318],{"className":3317},[821,822,823,824],[574,3319,3321,3324],{"className":3320},[633,824],[574,3322,695],{"className":3323},[633,824],[574,3325,601],{"className":3326},[633,634,824],[574,3328],{"className":3329,"style":677},[628],[574,3331,729],{"className":3332},[681],[574,3334],{"className":3335,"style":677},[628],[574,3337,3339,3342],{"className":3338},[615],[574,3340],{"className":3341,"style":1235},[619],[574,3343,2357],{"className":3344},[633]," nur die rechte Seite logarithmieren. Es werden immer ",[569,3347,3348],{},"beide"," Seiten logarithmiert.",[1174,3351,3352,3382,3383,3489,3490,3560,3561,1301],{},[574,3353,3355,3370],{"className":3354},[577],[574,3356,3358],{"className":3357},[581],[583,3359,3360],{"xmlns":585},[587,3361,3362,3368],{},[590,3363,3364,3366],{},[593,3365,550],{},[596,3367,598],{},[603,3369,1708],{"encoding":605},[574,3371,3373],{"className":3372,"ariaHidden":611},[610],[574,3374,3376,3379],{"className":3375},[615],[574,3377],{"className":3378,"style":620},[619],[574,3380,550],{"className":3381},[624]," mit ",[574,3384,3386,3415],{"className":3385},[577],[574,3387,3389],{"className":3388},[581],[583,3390,3391],{"xmlns":585},[587,3392,3393,3412],{},[590,3394,3395,3398,3400,3402],{},[593,3396,3397],{},"lg",[596,3399,598],{},[596,3401,729],{},[731,3403,3404,3410],{},[590,3405,3406,3408],{},[593,3407,737],{},[596,3409,598],{},[655,3411,2113],{},[603,3413,3414],{"encoding":605},"\\lg = \\log_{10}",[574,3416,3418,3438],{"className":3417,"ariaHidden":611},[610],[574,3419,3421,3424,3429,3432,3435],{"className":3420},[615],[574,3422],{"className":3423,"style":2219},[619],[574,3425,995,3427],{"className":3426},[624],[574,3428,791],{"style":790},[574,3430],{"className":3431,"style":677},[628],[574,3433,729],{"className":3434},[681],[574,3436],{"className":3437,"style":677},[628],[574,3439,3441,3444],{"className":3440},[615],[574,3442],{"className":3443,"style":780},[619],[574,3445,3447,3452],{"className":3446},[624],[574,3448,787,3450],{"className":3449},[624],[574,3451,791],{"style":790},[574,3453,3455],{"className":3454},[795],[574,3456,3458,3481],{"className":3457},[799,800],[574,3459,3461,3478],{"className":3460},[804],[574,3462,3464],{"className":3463,"style":2745},[808],[574,3465,3466,3469],{"style":812},[574,3467],{"className":3468,"style":817},[816],[574,3470,3472],{"className":3471},[821,822,823,824],[574,3473,3475],{"className":3474},[633,824],[574,3476,2113],{"className":3477},[633,824],[574,3479,832],{"className":3480},[831],[574,3482,3484],{"className":3483},[804],[574,3485,3487],{"className":3486,"style":839},[808],[574,3488],{}," verwechseln: ",[574,3491,3493,3515],{"className":3492},[577],[574,3494,3496],{"className":3495},[581],[583,3497,3498],{"xmlns":585},[587,3499,3500,3512],{},[590,3501,3502,3504,3506,3508,3510],{},[593,3503,550],{},[596,3505,598],{},[655,3507,2113],{},[596,3509,653],{},[655,3511,2118],{},[603,3513,3514],{"encoding":605},"\\ln 10 \\approx 2{,}3026",[574,3516,3518,3542],{"className":3517,"ariaHidden":611},[610],[574,3519,3521,3524,3527,3530,3533,3536,3539],{"className":3520},[615],[574,3522],{"className":3523,"style":620},[619],[574,3525,550],{"className":3526},[624],[574,3528],{"className":3529,"style":629},[628],[574,3531,2113],{"className":3532},[633],[574,3534],{"className":3535,"style":677},[628],[574,3537,653],{"className":3538},[681],[574,3540],{"className":3541,"style":677},[628],[574,3543,3545,3548,3551,3557],{"className":3544},[615],[574,3546],{"className":3547,"style":691},[619],[574,3549,695],{"className":3550},[633],[574,3552,3554],{"className":3553},[633],[574,3555,703],{"className":3556},[702],[574,3558,2268],{"className":3559},[633],", aber ",[574,3562,3564,3586],{"className":3563},[577],[574,3565,3567],{"className":3566},[581],[583,3568,3569],{"xmlns":585},[587,3570,3571,3583],{},[590,3572,3573,3575,3577,3579,3581],{},[593,3574,3397],{},[596,3576,598],{},[655,3578,2113],{},[596,3580,729],{},[655,3582,1194],{},[603,3584,3585],{"encoding":605},"\\lg 10 = 1",[574,3587,3589,3615],{"className":3588,"ariaHidden":611},[610],[574,3590,3592,3595,3600,3603,3606,3609,3612],{"className":3591},[615],[574,3593],{"className":3594,"style":2219},[619],[574,3596,995,3598],{"className":3597},[624],[574,3599,791],{"style":790},[574,3601],{"className":3602,"style":629},[628],[574,3604,2113],{"className":3605},[633],[574,3607],{"className":3608,"style":677},[628],[574,3610,729],{"className":3611},[681],[574,3613],{"className":3614,"style":677},[628],[574,3616,3618,3621],{"className":3617},[615],[574,3619],{"className":3620,"style":1235},[619],[574,3622,1194],{"className":3623},[633],{"title":3625,"searchDepth":3626,"depth":3626,"links":3627},"",2,[3628,3629,3630],{"id":562,"depth":3626,"text":563},{"id":2065,"depth":3626,"text":2066},{"id":2893,"depth":3626,"text":2894},[3632,3634,3636,3638,3640,3642,3645,3647,3650,3653,3656,3659],{"lhs":3633,"rhs":601},"exp(log(x))",{"lhs":3635,"rhs":601},"log(exp(x))",{"lhs":3637,"rhs":901},"log(1)",{"lhs":3639,"rhs":1194},"log(exp(1))",{"lhs":3641,"rhs":2096},"log(exp(3))",{"lhs":3643,"rhs":3644},"round(log(10), 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mathnormal\">x\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\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md","natural-log","\\ln x = y \\;\\Leftrightarrow\\; e^y = x","\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>\u003Cmi>ln\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>y\u003C\u002Fmi>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmo>⇔\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmsup>\u003Cmi>e\u003C\u002Fmi>\u003Cmi>y\u003C\u002Fmi>\u003C\u002Fmsup>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\ln x = y \\;\\Leftrightarrow\\; e^y = x\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>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\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:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\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>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.7144em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">e\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7144em;\">\u003Cspan style=\"top:-3.113em;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:0.0359em;\">y\u003C\u002Fspan>\u003C\u002Fspan>\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:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","upper",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Falgebra\u002Fnatuerlicher-logarithmus",[],[3675,3676,3677,3678],"log-definition","natural-exponential-function","log-function","log-change-of-base",{"title":548,"description":3625},"de\u002Fformulas\u002Fmath\u002Falgebra\u002Fnatuerlicher-logarithmus","Der natürliche Logarithmus ln x ist der Exponent, mit dem man e potenzieren muss, um x zu erhalten.",[3683,550,3684,3685,3686],"logarithmus","eulersche zahl","e-funktion","umkehrfunktion",[3688,3691,3694],{"symbol":601,"meaning":3689,"symbolHtml":3690},"Numerus, eine positive Zahl","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>x\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation 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style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":650,"meaning":3695,"symbolHtml":3696},"Eulersche Zahl, e ≈ 2,71828","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>e\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">e\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:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord 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