# 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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