Logarithms invert exponentials

The func­tion \(\log_b(\cdot)\) in­verts the func­tion \(b^{(\cdot)}.\) In other words, \(\log_b(n) = x\) im­plies that \(b^x = n,\) so \(\log_b(b^x)=x\) and \(b^{\log_b(n)}=n.\) (For ex­am­ple, \(\log_2(2^3) = 3\) and \(2^{\log_2(8)} = 8.\)) Thus, log­a­r­ithms give us tools for an­a­lyz­ing any­thing that grows ex­po­nen­tially. If a pop­u­la­tion of bac­te­ria grows ex­po­nen­tially, then log­a­r­ithms can be used to an­swer ques­tions about how long it will take the pop­u­la­tion to reach a cer­tain size. If your wealth is ac­cu­mu­lat­ing in­ter­est, log­a­r­ithms can be used to ask how long it will take un­til you have a cer­tain amount of wealth. (TODO)

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