Logarithmic+Functions

by **[|ViaMoi]**



toc =**__Definition__**= log //b//  //x// = n means b  //n//  = //x//. Logarithms replace a geometric series with an arithmetic series.

= **__Log Rules__** = = = We can observe that in any base, the logarithm of 1 is 0. log //b// 1 = 0

In any base, the logarithm of the base itself is 1. log //b// b = 1

The following is an important formal rule, valid for any base b: log //b// //b// //x// = //x// =**__Common logarithms__**= The system of common logarithms has 10 as its base. When the base is not indicated, log 100 = 2 then the system of common logarithms -- base 10 -- is implied. Here are the powers of 10 and their logarithms: 1000 ||  || __1__ 100 ||   || __1__ 10 ||   || 1 ||   || 10 ||   || 100 ||   || 1000 ||   || 10,000 || =**__Natural logarithms__**= The system of natural logarithms has the number called "//e//" as its base. =**__The three laws of logarithms__**= "//The logarithm of a product is equal to the sum of the logarithms of each factor.//" //y// || = log //b// //x// − log //b// //y// || "//The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator.//"  "//The logarithm of a power of// x //is equal to the exponent of that power times the logarithm of// x//.//"
 * **Powers of 10:** ||  || __1__
 * **Logarithms:** ||  || −3 ||   || −2 ||   || −1 ||   || 0 ||   || 1 ||   || 2 ||   || 3 ||   || 4 ||
 * **Logarithms:** ||  || −3 ||   || −2 ||   || −1 ||   || 0 ||   || 1 ||   || 2 ||   || 3 ||   || 4 ||
 * 1**. log //b(// //xy)// = log //b// //x// + log //b// //y//
 * **2**. log //b// || __//x//__
 * 3**. log //b// //x// //^n// = n log //b// //x//