Logarithms invert exponentials

The function ${\log }_b(\cdot )$ inverts the function $b^{(\cdot )}.$ In other words, ${\log }_b\left(n\right)=x$ implies that $b^x=n,$ so ${\log }_b\left(b^x\right)=x$ and $b^{{\log }_b\left(n\right)}=n.$ (For example, ${\log }_2\left(2^3\right)=3$ and $2^{{\log }_2\left(8\right)}=8.$) 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)