Chapter 1.6, Problem 383E

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.

383. Use the identity $\text{In}\left(x\right)={\displaystyle {\int }_1^x\frac{dt}{t}}$ to derive the identity $\text{In}\left(\frac{1}{x}\right)=-\text{In}x$ .