Question

Expanding Logarithmic Expressions In Exercise, use the properties of logarithms to rewrite the expression as a sum, difference, or multiple of logarithms.
In (x^2 - 1)^1/3

Answer 1

Answer:  $\dfrac{1}{3}[\ln (x+1)+\ln(x-1)]$

Step-by-step explanation:

Properties of logarithm :

1. $\log (ab)= \log a+\log b$
2. $\log(\dfrac{a}{b})=\log a-\log b$
3. $\log a^n= n\log a$

The given expression in terms of Natural log : $\ln (x^2 - 1)^{\frac{1}{3}}$

This will become $\dfrac{1}{3}\ln (x^2 - 1)$      [ By using Property (3)]

$=\dfrac{1}{3}\ln (x^2-1^2)$

$=\dfrac{1}{3}\ln ((x+1)(x-1))$   [$\because a^2-b^2=(a+b)(a-b)$]

$=\dfrac{1}{3}[\ln (x+1)+\ln(x-1)]$   [ By using Property (1)]

Hence, the simplified expression becomes $\dfrac{1}{3}[\ln (x+1)+\ln(x-1)]$ .