Q. 77 - Calculus

Question

Use the previous two exercises to prove that for any $a,b>0$,  $\ln \left(\frac{a}{b}\right)=\ln a-\ln b$.

Step-by-Step Solution

Answer

$\ln \left(\frac{a}{b}\right)=\ln a-\ln b$ for any $a,b>0$.

Step 1. Given information

We have to prove that $\ln \left(\frac{a}{b}\right)=\ln a-\ln b$ for any $a,b>0$.

Step 2. Proof of the question.

$$\begin{gathered} \ln \left(\frac{a}{b}\right) \\ =\ln \left(a{b}^{-1}\right) \\ =\ln a+\ln \left(b^{-1}\right) \\ =\ln a-\ln b \end{gathered}$$

Therefore, it is proved that $\ln \left(\frac{a}{b}\right)=\ln a-\ln b$ for any $a,b>0$.