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)1/3

Answer 1

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

Explanation:

Consider the given expression is

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

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

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 equate form of given expression is
`\ln x+(1)/(3)\ln (x^2 + 1)` .

Answer 2

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

Explanation:

We are given the following expression in the question

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

Logarithmic Properties:

`\log (ab) = \log a + \log b\\\\\log (a)/(b) = \log a - \log b\\\\\log (a^b) = b\log a`

We have to simplify the given expression

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

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