Question

Condensing Logarithmic Expressions In Exercise, use the properties of logarithms to rewrite the expression as the logarithm of a single quantity.

1/3 In(x^2 - 6) - 2 In (3x + 2)

Answer 1

`\ln (\frac{(x^2-6)^{(1)/(3)}}{(3x+2)^2})`

Explanation:

The given expression is

`(1)/(3)\ln (x^2-6)-2\ln (3x+2)`

We need to rewrite the expression as the logarithm of a single quantity.

Using the properties of logarithm we get

`\ln (x^2-6)^{(1)/(3)}-\ln (3x+2)^2`
`[\because \log a^b = b\log a]`

`\ln (\frac{(x^2-6)^{(1)/(3)}}{(3x+2)^2})`
`[\because \log a-log b = \log ((a)/(b))]`

Therefore, the simplified form of the given expression is
`\ln (\frac{(x^2-6)^{(1)/(3)}}{(3x+2)^2})`.