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

Answer:

$\ln x+\frac{1}{3}\ln (x^2+1)$

Step-by-step 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.

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

Using the properties of logarithm we get

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

$\ln x+\frac{1}{3}\ln (x^2+1)$        $[\because \ln (a^b)=b\ln a]$

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