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\\cdot \\log_c a = \\log_c b\n\\quad\\Rightarrow\\quad\nx = \\frac{\\log_c b}{\\log_c a}",[567,1135,1137,1158,1221,1291,1309],{"className":1136,"ariaHidden":597},[596],[567,1138,1140,1144,1147,1151,1155],{"className":1139},[601],[567,1141],{"className":1142,"style":1143},[605],"height:0.4445em;",[567,1145,806],{"className":1146},[610,611],[567,1148],{"className":1149,"style":1150},[843],"margin-right:0.2222em;",[567,1152,1061],{"className":1153},[1154],"mbin",[567,1156],{"className":1157,"style":1150},[843],[567,1159,1161,1164,1206,1209,1212,1215,1218],{"className":1160},[601],[567,1162],{"className":1163,"style":858},[605],[567,1165,1167,1172],{"className":1166},[683],[567,1168,865,1170],{"className":1169},[683],[567,1171,752],{"style":751},[567,1173,1175],{"className":1174},[871],[567,1176,1178,1198],{"className":1177},[875,876],[567,1179,1181,1195],{"className":1180},[880],[567,1182,1184],{"className":1183,"style":885},[884],[567,1185,1186,1189],{"style":888},[567,1187],{"className":1188,"style":893},[892],[567,1190,1192],{"className":1191},[897,898,899,900],[567,1193,632],{"className":1194},[610,611,900],[567,1196,908],{"className":1197},[907],[567,1199,1201],{"className":1200},[880],[567,1202,1204],{"className":1203,"style":915},[884],[567,1205],{},[567,1207],{"className":1208,"style":921},[843],[567,1210,588],{"className":1211},[610,611],[567,1213],{"className":1214,"style":844},[843],[567,1216,809],{"className":1217},[848],[567,1219],{"className":1220,"style":844},[843],[567,1222,1224,1227,1269,1272,1275,1279,1282,1285,1288],{"className":1223},[601],[567,1225],{"className":1226,"style":858},[605],[567,1228,1230,1235],{"className":1229},[683],[567,1231,865,1233],{"className":1232},[683],[567,1234,752],{"style":751},[567,1236,1238],{"className":1237},[871],[567,1239,1241,1261],{"className":1240},[875,876],[567,1242,1244,1258],{"className":1243},[880],[567,1245,1247],{"className":1246,"style":885},[884],[567,1248,1249,1252],{"style":888},[567,1250],{"className":1251,"style":893},[892],[567,1253,1255],{"className":1254},[897,898,899,900],[567,1256,632],{"className":1257},[610,611,900],[567,1259,908],{"className":1260},[907],[567,1262,1264],{"className":1263},[880],[567,1265,1267],{"className":1266,"style":915},[884],[567,1268],{},[567,1270],{"className":1271,"style":921},[843],[567,1273,824],{"className":1274},[610,611],[567,1276],{"className":1277,"style":1278},[843],"margin-right:1em;",[567,1280],{"className":1281,"style":844},[843],[567,1283,1093],{"className":1284},[848],[567,1286],{"className":1287,"style":1278},[843],[567,1289],{"className":1290,"style":844},[843],[567,1292,1294,1297,1300,1303,1306],{"className":1293},[601],[567,1295],{"className":1296,"style":606},[605],[567,1298,806],{"className":1299},[610,611],[567,1301],{"className":1302,"style":844},[843],[567,1304,809],{"className":1305},[848],[567,1307],{"className":1308,"style":844},[843],[567,1310,1312,1316],{"className":1311},[601],[567,1313],{"className":1314,"style":1315},[605],"height:2.3016em;vertical-align:-0.9301em;",[567,1317,1319,1324,1475],{"className":1318},[610],[567,1320],{"className":1321},[1322,1323],"mopen","nulldelimiter",[567,1325,1327],{"className":1326},[1101],[567,1328,1330,1466],{"className":1329},[875,876],[567,1331,1333,1463],{"className":1332},[880],[567,1334,1337,1395,1406],{"className":1335,"style":1336},[884],"height:1.3714em;",[567,1338,1340,1344],{"style":1339},"top:-2.314em;",[567,1341],{"className":1342,"style":1343},[892],"height:3em;",[567,1345,1347,1389,1392],{"className":1346},[610],[567,1348,1350,1355],{"className":1349},[683],[567,1351,865,1353],{"className":1352},[683],[567,1354,752],{"style":751},[567,1356,1358],{"className":1357},[871],[567,1359,1361,1381],{"className":1360},[875,876],[567,1362,1364,1378],{"className":1363},[880],[567,1365,1367],{"className":1366,"style":885},[884],[567,1368,1369,1372],{"style":888},[567,1370],{"className":1371,"style":893},[892],[567,1373,1375],{"className":1374},[897,898,899,900],[567,1376,632],{"className":1377},[610,611,900],[567,1379,908],{"className":1380},[907],[567,1382,1384],{"className":1383},[880],[567,1385,1387],{"className":1386,"style":915},[884],[567,1388],{},[567,1390],{"className":1391,"style":921},[843],[567,1393,588],{"className":1394},[610,611],[567,1396,1398,1401],{"style":1397},"top:-3.23em;",[567,1399],{"className":1400,"style":1343},[892],[567,1402],{"className":1403,"style":1405},[1404],"frac-line","border-bottom-width:0.04em;",[567,1407,1409,1412],{"style":1408},"top:-3.677em;",[567,1410],{"className":1411,"style":1343},[892],[567,1413,1415,1457,1460],{"className":1414},[610],[567,1416,1418,1423],{"className":1417},[683],[567,1419,865,1421],{"className":1420},[683],[567,1422,752],{"style":751},[567,1424,1426],{"className":1425},[871],[567,1427,1429,1449],{"className":1428},[875,876],[567,1430,1432,1446],{"className":1431},[880],[567,1433,1435],{"className":1434,"style":885},[884],[567,1436,1437,1440],{"style":888},[567,1438],{"className":1439,"style":893},[892],[567,1441,1443],{"className":1442},[897,898,899,900],[567,1444,632],{"className":1445},[610,611,900],[567,1447,908],{"className":1448},[907],[567,1450,1452],{"className":1451},[880],[567,1453,1455],{"className":1454,"style":915},[884],[567,1456],{},[567,1458],{"className":1459,"style":921},[843],[567,1461,824],{"className":1462},[610,611],[567,1464,908],{"className":1465},[907],[567,1467,1469],{"className":1468},[880],[567,1470,1473],{"className":1471,"style":1472},[884],"height:0.9301em;",[567,1474],{},[567,1476],{"className":1477},[1478,1323],"mclose",[558,1480,1482],{"id":1481},"beispiele","Beispiele",[567,1484,1486],{"className":1485},[1043],[567,1487,1489,1550],{"className":1488},[570],[567,1490,1492],{"className":1491},[574],[576,1493,1494],{"xmlns":578,"display":1052},[580,1495,1496,1547],{},[583,1497,1498,1509,1511,1513,1531,1534,1542,1544],{},[811,1499,1500,1506],{},[583,1501,1502,1504],{},[586,1503,817],{},[664,1505,666],{},[766,1507,1508],{},"2",[766,1510,768],{},[664,1512,809],{},[1101,1514,1515,1523],{},[583,1516,1517,1519,1521],{},[586,1518,662],{},[664,1520,666],{},[766,1522,768],{},[583,1524,1525,1527,1529],{},[586,1526,662],{},[664,1528,666],{},[766,1530,1508],{},[664,1532,1533],{},"≈",[1101,1535,1536,1539],{},[766,1537,1538],{},"2,3026",[766,1540,1541],{},"0,6931",[664,1543,1533],{},[766,1545,1546],{},"3,32",[590,1548,1549],{"encoding":592},"\\log_2 10 = \\frac{\\ln 10}{\\ln 2} \\approx \\frac{2{,}3026}{0{,}6931} \\approx 3{,}32",[567,1551,1553,1617,1708,1811],{"className":1552,"ariaHidden":597},[596],[567,1554,1556,1559,1602,1605,1608,1611,1614],{"className":1555},[601],[567,1557],{"className":1558,"style":858},[605],[567,1560,1562,1567],{"className":1561},[683],[567,1563,865,1565],{"className":1564},[683],[567,1566,752],{"style":751},[567,1568,1570],{"className":1569},[871],[567,1571,1573,1594],{"className":1572},[875,876],[567,1574,1576,1591],{"className":1575},[880],[567,1577,1580],{"className":1578,"style":1579},[884],"height:0.207em;",[567,1581,1582,1585],{"style":888},[567,1583],{"className":1584,"style":893},[892],[567,1586,1588],{"className":1587},[897,898,899,900],[567,1589,1508],{"className":1590},[610,900],[567,1592,908],{"className":1593},[907],[567,1595,1597],{"className":1596},[880],[567,1598,1600],{"className":1599,"style":915},[884],[567,1601],{},[567,1603],{"className":1604,"style":921},[843],[567,1606,768],{"className":1607},[610],[567,1609],{"className":1610,"style":844},[843],[567,1612,809],{"className":1613},[848],[567,1615],{"className":1616,"style":844},[843],[567,1618,1620,1624,1699,1702,1705],{"className":1619},[601],[567,1621],{"className":1622,"style":1623},[605],"height:2.0574em;vertical-align:-0.686em;",[567,1625,1627,1630,1696],{"className":1626},[610],[567,1628],{"className":1629},[1322,1323],[567,1631,1633],{"className":1632},[1101],[567,1634,1636,1687],{"className":1635},[875,876],[567,1637,1639,1684],{"className":1638},[880],[567,1640,1642,1659,1667],{"className":1641,"style":1336},[884],[567,1643,1644,1647],{"style":1339},[567,1645],{"className":1646,"style":1343},[892],[567,1648,1650,1653,1656],{"className":1649},[610],[567,1651,662],{"className":1652},[683],[567,1654],{"className":1655,"style":921},[843],[567,1657,1508],{"className":1658},[610],[567,1660,1661,1664],{"style":1397},[567,1662],{"className":1663,"style":1343},[892],[567,1665],{"className":1666,"style":1405},[1404],[567,1668,1669,1672],{"style":1408},[567,1670],{"className":1671,"style":1343},[892],[567,1673,1675,1678,1681],{"className":1674},[610],[567,1676,662],{"className":1677},[683],[567,1679],{"className":1680,"style":921},[843],[567,1682,768],{"className":1683},[610],[567,1685,908],{"className":1686},[907],[567,1688,1690],{"className":1689},[880],[567,1691,1694],{"className":1692,"style":1693},[884],"height:0.686em;",[567,1695],{},[567,1697],{"className":1698},[1478,1323],[567,1700],{"className":1701,"style":844},[843],[567,1703,1533],{"className":1704},[848],[567,1706],{"className":1707,"style":844},[843],[567,1709,1711,1715,1802,1805,1808],{"className":1710},[601],[567,1712],{"className":1713,"style":1714},[605],"height:2.2019em;vertical-align:-0.8804em;",[567,1716,1718,1721,1799],{"className":1717},[610],[567,1719],{"className":1720},[1322,1323],[567,1722,1724],{"className":1723},[1101],[567,1725,1727,1790],{"className":1726},[875,876],[567,1728,1730,1787],{"className":1729},[880],[567,1731,1734,1758,1766],{"className":1732,"style":1733},[884],"height:1.3214em;",[567,1735,1736,1739],{"style":1339},[567,1737],{"className":1738,"style":1343},[892],[567,1740,1742,1746,1754],{"className":1741},[610],[567,1743,1745],{"className":1744},[610],"0",[567,1747,1749],{"className":1748},[610],[567,1750,1753],{"className":1751},[1752],"mpunct",",",[567,1755,1757],{"className":1756},[610],"6931",[567,1759,1760,1763],{"style":1397},[567,1761],{"className":1762,"style":1343},[892],[567,1764],{"className":1765,"style":1405},[1404],[567,1767,1768,1771],{"style":1408},[567,1769],{"className":1770,"style":1343},[892],[567,1772,1774,1777,1783],{"className":1773},[610],[567,1775,1508],{"className":1776},[610],[567,1778,1780],{"className":1779},[610],[567,1781,1753],{"className":1782},[1752],[567,1784,1786],{"className":1785},[610],"3026",[567,1788,908],{"className":1789},[907],[567,1791,1793],{"className":1792},[880],[567,1794,1797],{"className":1795,"style":1796},[884],"height:0.8804em;",[567,1798],{},[567,1800],{"className":1801},[1478,1323],[567,1803],{"className":1804,"style":844},[843],[567,1806,1533],{"className":1807},[848],[567,1809],{"className":1810,"style":844},[843],[567,1812,1814,1818,1822,1828],{"className":1813},[601],[567,1815],{"className":1816,"style":1817},[605],"height:0.8389em;vertical-align:-0.1944em;",[567,1819,1821],{"className":1820},[610],"3",[567,1823,1825],{"className":1824},[610],[567,1826,1753],{"className":1827},[1752],[567,1829,1831],{"className":1830},[610],"32",[567,1833,1835],{"className":1834},[1043],[567,1836,1838,1888],{"className":1837},[570],[567,1839,1841],{"className":1840},[574],[576,1842,1843],{"xmlns":578,"display":1052},[580,1844,1845,1885],{},[583,1846,1847,1858,1861,1863,1881,1883],{},[811,1848,1849,1855],{},[583,1850,1851,1853],{},[586,1852,817],{},[664,1854,666],{},[766,1856,1857],{},"5",[766,1859,1860],{},"125",[664,1862,809],{},[1101,1864,1865,1873],{},[583,1866,1867,1869,1871],{},[586,1868,729],{},[664,1870,666],{},[766,1872,1860],{},[583,1874,1875,1877,1879],{},[586,1876,729],{},[664,1878,666],{},[766,1880,1857],{},[664,1882,809],{},[766,1884,1821],{},[590,1886,1887],{"encoding":592},"\\log_5 125 = \\frac{\\lg 125}{\\lg 5} = 3",[567,1889,1891,1954,2048],{"className":1890,"ariaHidden":597},[596],[567,1892,1894,1897,1939,1942,1945,1948,1951],{"className":1893},[601],[567,1895],{"className":1896,"style":858},[605],[567,1898,1900,1905],{"className":1899},[683],[567,1901,865,1903],{"className":1902},[683],[567,1904,752],{"style":751},[567,1906,1908],{"className":1907},[871],[567,1909,1911,1931],{"className":1910},[875,876],[567,1912,1914,1928],{"className":1913},[880],[567,1915,1917],{"className":1916,"style":1579},[884],[567,1918,1919,1922],{"style":888},[567,1920],{"className":1921,"style":893},[892],[567,1923,1925],{"className":1924},[897,898,899,900],[567,1926,1857],{"className":1927},[610,900],[567,1929,908],{"className":1930},[907],[567,1932,1934],{"className":1933},[880],[567,1935,1937],{"className":1936,"style":915},[884],[567,1938],{},[567,1940],{"className":1941,"style":921},[843],[567,1943,1860],{"className":1944},[610],[567,1946],{"className":1947,"style":844},[843],[567,1949,809],{"className":1950},[848],[567,1952],{"className":1953,"style":844},[843],[567,1955,1957,1961,2039,2042,2045],{"className":1956},[601],[567,1958],{"className":1959,"style":1960},[605],"height:2.2519em;vertical-align:-0.8804em;",[567,1962,1964,1967,2036],{"className":1963},[610],[567,1965],{"className":1966},[1322,1323],[567,1968,1970],{"className":1969},[1101],[567,1971,1973,2028],{"className":1972},[875,876],[567,1974,1976,2025],{"className":1975},[880],[567,1977,1979,1998,2006],{"className":1978,"style":1336},[884],[567,1980,1981,1984],{"style":1339},[567,1982],{"className":1983,"style":1343},[892],[567,1985,1987,1992,1995],{"className":1986},[610],[567,1988,748,1990],{"className":1989},[683],[567,1991,752],{"style":751},[567,1993],{"className":1994,"style":921},[843],[567,1996,1857],{"className":1997},[610],[567,1999,2000,2003],{"style":1397},[567,2001],{"className":2002,"style":1343},[892],[567,2004],{"className":2005,"style":1405},[1404],[567,2007,2008,2011],{"style":1408},[567,2009],{"className":2010,"style":1343},[892],[567,2012,2014,2019,2022],{"className":2013},[610],[567,2015,748,2017],{"className":2016},[683],[567,2018,752],{"style":751},[567,2020],{"className":2021,"style":921},[843],[567,2023,1860],{"className":2024},[610],[567,2026,908],{"className":2027},[907],[567,2029,2031],{"className":2030},[880],[567,2032,2034],{"className":2033,"style":1796},[884],[567,2035],{},[567,2037],{"className":2038},[1478,1323],[567,2040],{"className":2041,"style":844},[843],[567,2043,809],{"className":2044},[848],[567,2046],{"className":2047,"style":844},[843],[567,2049,2051,2054],{"className":2050},[601],[567,2052],{"className":2053,"style":780},[605],[567,2055,1821],{"className":2056},[610],[558,2058,2060],{"id":2059},"typische-fehler","Typische Fehler",[2062,2063,2064,2126],"ul",{},[2065,2066,2067,2068,2096,2097,2125],"li",{},"Zähler und Nenner vertauschen: Der Numerus ",[567,2069,2071,2084],{"className":2070},[570],[567,2072,2074],{"className":2073},[574],[576,2075,2076],{"xmlns":578},[580,2077,2078,2082],{},[583,2079,2080],{},[586,2081,824],{},[590,2083,824],{"encoding":592},[567,2085,2087],{"className":2086,"ariaHidden":597},[596],[567,2088,2090,2093],{"className":2089},[601],[567,2091],{"className":2092,"style":679},[605],[567,2094,824],{"className":2095},[610,611]," gehört in den Zähler, die alte Basis ",[567,2098,2100,2113],{"className":2099},[570],[567,2101,2103],{"className":2102},[574],[576,2104,2105],{"xmlns":578},[580,2106,2107,2111],{},[583,2108,2109],{},[586,2110,588],{},[590,2112,588],{"encoding":592},[567,2114,2116],{"className":2115,"ariaHidden":597},[596],[567,2117,2119,2122],{"className":2118},[601],[567,2120],{"className":2121,"style":606},[605],[567,2123,588],{"className":2124},[610,611]," in den Nenner.",[2065,2127,2128,2129,2413],{},"Den Bruch zu einem Logarithmus „kürzen“: ",[567,2130,2132,2176],{"className":2131},[570],[567,2133,2135],{"className":2134},[574],[576,2136,2137],{"xmlns":578},[580,2138,2139,2173],{},[583,2140,2141,2159,2163,2165,2167],{},[1101,2142,2143,2151],{},[583,2144,2145,2147,2149],{},[586,2146,662],{},[664,2148,666],{},[586,2150,824],{},[583,2152,2153,2155,2157],{},[586,2154,662],{},[664,2156,666],{},[586,2158,588],{},[664,2160,2162],{"mathvariant":2161},"normal","≠",[586,2164,662],{},[664,2166,666],{},[1101,2168,2169,2171],{},[586,2170,824],{},[586,2172,588],{},[590,2174,2175],{"encoding":592},"\\frac{\\ln b}{\\ln a} \\neq \\ln 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a)","log(b) \u002F log(a)",{"lhs":2424,"rhs":2427},"log10(b) \u002F log10(a)",{"lhs":2429,"rhs":2430},"log(10) \u002F log(2)","log(10, 2)",{"lhs":2432,"rhs":1821},"log10(125) \u002F log10(5)","a, c > 0,\\ a, c \\neq 1,\\ b > 0","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>a\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>c\u003C\u002Fmi>\u003Cmo>>\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a, c > 0,\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\">a\u003C\u002Fspan>\u003Cspan 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mathvariant=\"normal\">≠\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a, c \\neq 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:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">\u003Cspan class=\"mrel\">\u003Cspan class=\"mord katex-vbox\">\u003Cspan class=\"katex-thinbox\">\u003Cspan class=\"rlap\">\u003Cspan class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-inner\">\u003Cspan class=\"mord\">\u003Cspan class=\"mrel\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-fix\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace nobreak\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>b\u003C\u002Fmi>\u003Cmo>>\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">b > 0\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:-0.0391em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">b\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","log-change-of-base","\\log_a b = \\frac{\\log_c b}{\\log_c a} = \\frac{\\ln b}{\\ln a}","\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>\u003Cmrow>\u003Cmi>log\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmi>a\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmi>b\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmfrac>\u003Cmrow>\u003Cmsub>\u003Cmrow>\u003Cmi>log\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmi>c\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmi>b\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cmrow>\u003Cmsub>\u003Cmrow>\u003Cmi>log\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmi>c\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmi>a\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmfrac>\u003Cmo>=\u003C\u002Fmo>\u003Cmfrac>\u003Cmrow>\u003Cmi>ln\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>b\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cmrow>\u003Cmi>ln\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>a\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmfrac>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\log_a b = \\frac{\\log_c b}{\\log_c a} = \\frac{\\ln b}{\\ln a}\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:0.9386em;vertical-align:-0.2441em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mop\">lo\u003Cspan style=\"margin-right:0.0139em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.0573em;\">\u003Cspan style=\"top:-2.4559em;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\">a\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:0.2441em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">b\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:2.3016em;vertical-align:-0.9301em;\">\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\">\u003Cspan class=\"mop\">lo\u003Cspan style=\"margin-right:0.0139em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.0573em;\">\u003Cspan style=\"top:-2.4559em;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\">c\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:0.2441em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\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\">\u003Cspan class=\"mop\">lo\u003Cspan style=\"margin-right:0.0139em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.0573em;\">\u003Cspan style=\"top:-2.4559em;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\">c\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan 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