Q. 73 - Calculus

Question

Prove that $\ln x$ is zero if $x=1$, negative if $0<x<1$, and positive if $x>1$.

Step-by-Step Solution

Answer

$\ln x$ is zero if $x=1$, negative if $0<x<1$, and positive if $x>1$.

Step 1. Given information

We have to prove that $\ln x$ is zero if $x=1$, negative if $0<x<1$, and positive if $x>1$.

Step 2. Proof of the question.

$\ln x$ can be written as ${\int }_1^x\frac{1}{t}dt$

For $x=1$,

$$\begin{gathered} \ln 1={\int }_1^1\frac{1}{t}dt \\ =0 \end{gathered}$$

For $0<x<1$,

$$\begin{gathered} \ln x={\int }_1^x\frac{1}{t}dt \\ =-{\int }_x^1\frac{1}{t}dt \end{gathered}$$

So, $\ln x$ is negative.

Now for $x>1$,

$\ln x={\int }_1^x\frac{1}{t}dt$

So, $\ln x$ is positive.