Exercise 8.4.6

(a)

Explain why we know $e^x$ has an inverse function—let’s call it $\log <!--\; FUNCTION\; APPLICATION\; -->x$—defined on the strictly positive real numbers and satisfying

(i)

$\log <!--\; FUNCTION\; APPLICATION\; -->(e^y)=y$ for all $y\in R$ and

(ii)

$e^{\log <!--\; FUNCTION\; APPLICATION\; -->x}=x$, for all $x>0$.

(b)

Prove ${(\log <!--\; FUNCTION\; APPLICATION\; -->x)}^{\prime }=1∕x.$ (See Exercise 5.2.12.)

(c)

Fix $y>0$ and differentiate $\log <!--\; FUNCTION\; APPLICATION\; -->(\mathit{xy})$ with respect to $x$. Conclude that

$$\log <!--\; FUNCTION\; APPLICATION\; -->(\mathit{xy})=\log <!--\; FUNCTION\; APPLICATION\; -->x+\log <!--\; FUNCTION\; APPLICATION\; -->y\text{ for all }x,y>0.$$

(d)

For $t>0$ and $n\in N$, $t^n$ has the usual interpretation as $t\cdot t\cdots t$ (n times). Show that

$$t^n=e^{n\log <!--\; FUNCTION\; APPLICATION\; -->t}\text{ for all }n\in N$$