$8\mathrm{.}3\mathrm{.}9$ Theorem The logarithm is a strictly increasing function $L$ with domain $\{xinR$:$x>0\}$ and range $R.$ The derivative of $L$ is given by $($vii$)L^{\text{'}}(x)=\frac{1}{x}$ for $x>0.$ The logarithm satisfies the functional equation $($viii$)L(xy)=L(x)+L(y)$ for $x>0,y>0.$ Moreover, we have $($ix$)L(1)=0$ and $L(e)=1,($x$)L(x^r)=rL(x)$ for $x>0,$rinQ. $($xi$)\underset{x0+}{lim}L(x)=-$ and $\underset{x}{lim}L(x)=.$