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

Answer:

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

Step-by-step explanation:

We are given the following expression in the question

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

Logarithmic Properties:

$\log (ab) = \log a + \log b\\\\\log \dfrac{a}{b} = \log a - \log b\\\\\log (a^b) = b\log a$

We have to simplify the given expression

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

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

Answer 2

Answer:

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

Step-by-step explanation:

Consider the given expression is

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

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

Using the properties of logarithm we get

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

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

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