Question

Prove in your own words the last part of Theorem 4.37: If we define $\ln \left(x\right)={\int }_1^{\infty }\frac{1}{t}dt$for $x>0$, then $\ln \left(x\right)$is one-to-one on $\left(0,\infty \right)$.

Short answer

We have proved the theorem.

Step-by-step solution

Step 1. Given Information.

The objective is to prove the last part of Theorem 4.37.

Step 2. The proof.

The following graph shows that the signed area under the graph of $f=\frac{1}{t}$and x-axis is,