Question

Write each expression as a single logarithm.

$\frac{1}{3}\log \left(x^3+1\right)+\frac{1}{2}\log \left(x^2+1\right)$

Short answer

The obtained result is$\log \left[{\left(x^3+1\right)}^{\frac{1}{3}}{\left(x^2+1\right)}^{\frac{1}{2}}\right]$.

Step-by-step solution

Step 1. Given Information.

Consider the given information.

$\frac{1}{3}\log \left(x^3+1\right)+\frac{1}{2}\log \left(x^2+1\right)$

Step 2. Calculation.

Apply the logarithm properties to combine the terms.

$\begin{array}{rcl}\frac{1}{3}\log \left(x^3+1\right)+\frac{1}{2}\log \left(x^2+1\right)& =& \log {\left(x^3+1\right)}^{\frac{1}{3}}+\log {\left(x^2+1\right)}^{\frac{1}{2}}\\ & =& \log \left[{\left(x^3+1\right)}^{\frac{1}{3}}{\left(x^2+1\right)}^{\frac{1}{2}}\right]\end{array}$