Q. 71 - Calculus

Question

Prove each statement in Exercises 71–78, using the new definition of $\ln x$as an integral and $e^x$ as the inverse of $\ln x$.

71. Prove that $\ln x$ is continuous and differentiable on its

entire domain $(0,\infty )$.

Step-by-Step Solution

Answer

$\ln x$ is continuous and differentiable on its entire domain $\left(0,\infty \right)$.

Step 1. Given information

We have to prove that $\ln x$ is continuous and differentiable on its entire domain $\left(0,\infty \right)$.

Step 2. Proof of the question

The graph of the function $\frac{1}{t}$ is continuous on $\left(0,\infty \right)$.

The Second Fundamental Theorem tells that $\ln x={\int }_1^x\frac{1}{t}dt$.

Since, $\frac{1}{t}$ is continuous and integrable so $\ln x$ is also continuous and differentiable on $\left(0,\infty \right)$.