# 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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