# 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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