Answer: [tex]\dfrac{1}{3}[\ln (x+1)+\ln(x-1)][/tex]
Step-by-step explanation:
Properties of logarithm :
- [tex]\log (ab)= \log a+\log b[/tex]
- [tex]\log(\dfrac{a}{b})=\log a-\log b[/tex]
- [tex]\log a^n= n\log a[/tex]
The given expression in terms of Natural log : [tex] \ln (x^2 - 1)^{\frac{1}{3}}[/tex]
This will become [tex]\dfrac{1}{3}\ln (x^2 - 1)[/tex] [ By using Property (3)]
[tex]=\dfrac{1}{3}\ln (x^2-1^2)[/tex]
[tex]=\dfrac{1}{3}\ln ((x+1)(x-1))[/tex] [[tex]\because a^2-b^2=(a+b)(a-b)[/tex]]
[tex]=\dfrac{1}{3}[\ln (x+1)+\ln(x-1)][/tex] [ By using Property (1)]
Hence, the simplified expression becomes [tex]\dfrac{1}{3}[\ln (x+1)+\ln(x-1)][/tex] .
