Chapter 1.6, Problem 385E

The following exercises are intended to derive the fundamental properties of the natural log starting from the definition $\text{In}\left(x\right)={\displaystyle {\int }_1^x\frac{dt}{t}}$ , using properties of the definite integral and making no further assumptions.

385. Use the identity $\text{In}x={\displaystyle {\int }_1^x\frac{dt}{x}}$ to show that In(x) is an increasing function of x on $\left[0,\infty \right)$ , and use the previous exercises to show that the range of In(x) is $\left(-\infty ,\infty \right)$ . Without any further assumptions, conclude that In(x) has an inverse function defined on $\left(-\infty ,\infty \right)$ .