The function [3nd $~$\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)