Logarithm

Since $f(x)=a^x$ ($0<a\ne1$) is a bijection from $R$ onto $(0,\infty)$ (§2.8.3), it has an inverse function, called the logarithmic function with base $a$ and written $\log_a(\cdot)$: if $f$ sends $x\mapsto y=a^x$, then $\log_a(\cdot)$ sends $y\mapsto x$. Equivalently, for $0<a\ne1$,

For instance, $3^4=81\Rightarrow\log_3(81)=4$: given the base $a$ and a target value $y$, the logarithm finds the exponent $x$ with $a^x=y$ -- exactly the tool needed for questions like 'how long does an investment take to reach a given amount?', and, since $\log$ turns multiplication into addition, for multiplying very large or very small numbers by hand. …