Chapter 5.8, Problem 82E

To determine

(a)

To prove:

${\displaystyle {\int }_a^{ab}\frac{1}{t}dt={\displaystyle {\int }_1^b\frac{1}{t}dt}}$

To determine

(b)

To prove:

$G\left(ab\right)=G\left(a\right)+G\left(b\right)$

To determine

(c)

To Show: $G\left(1\right)=0\text{ and }G\left(a^{-1}\right)=-G\left(a\right)$

To determine

(d)

To show:

$G\left(a^n\right)=nG\left(a\right)$ for all $a>0$ and integers n.

To determine

(e)

To show:

$G\left(a^{1/n}\right)=\frac{1}{n}G\left(a\right)$ for all $a>0$ and integers n.

To determine

(f)

To show:

$G\left(a^r\right)=rG\left(a\right)$ for all $a>0$ and rational numberr

To determine

(g)

To show:

G is increasing.

To determine

(h)

To show:

There exists a number a such that $G\left(a\right)>1$ .

To determine

(i)

To show:

$\underset{x\to \infty }{\lim }G\left(x\right)=\infty \text{ and }\underset{x\to 0^{+}}{\lim }G\left(x\right)=-\infty$.

To determine

(j)

To show:

There exists a unique number E such that $G\left(E\right)=1$ .

To determine

(k)

To show:

$G\left(E^r\right)=r$ for every rational number r.