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:
`(1)/(3)[\ln (x+1)+\ln(x-1)]`

Explanation:

Properties of logarithm :

1. 
   `\log (ab)= \log a+\log b`
2. 
   `\log((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)^{(1)/(3)}`

This will become
`(1)/(3)\ln (x^2 - 1)` [ By using Property (3)]

`=(1)/(3)\ln (x^2-1^2)`

`=(1)/(3)\ln ((x+1)(x-1))` [
`\because a^2-b^2=(a+b)(a-b)`]

`=(1)/(3)[\ln (x+1)+\ln(x-1)]` [ By using Property (1)]

Hence, the simplified expression becomes
`(1)/(3)[\ln (x+1)+\ln(x-1)]` .