Question

Expanding logarithmic Expression In Exercise,Use the properties of logarithms to rewrite the expression as a sum,difference,or multipal of logarithms.See example 3.

In(x x^2 + 1^1/2)

Answer 1

`\ln x+(1)/(3)\ln (x^2+1)`

Explanation:

Consider the given expression is

`\ln (x\sqrt[3]{x^2+1})`

We need to rewrite the expression as a sum,difference,or multiple of logarithms.

`\ln (x(x^2+1)^{(1)/(3)})`
`[\because \sqrt[n]{x}=x^{(1)/(n)}]`

Using the properties of logarithm we get

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

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

Therefore, the simplified form of the given expression is
`\ln x+(1)/(3)\ln (x^2+1)`.