The function 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)

Logarithms invert exponentials