Question

Write each expression as a single logarithm.

$2{\log }_a\left(5x^3\right)-\frac{1}{2}{\log }_a\left(2x+3\right)$

Short answer

The result is ${\log }_a\frac{25x^6}{\sqrt{2x+3}}$.

Step-by-step solution

Step 1. Given information.

Consider the given information.

$2{\log }_a\left(5x^3\right)-\frac{1}{2}{\log }_a\left(2x+3\right)$

Step 2. Calculation.

Apply the logarithm properties to combine the terms.

$\begin{array}{rcl}2{\log }_a\left(5x^3\right)-\frac{1}{2}{\log }_a\left(2x+3\right)& =& {\log }_a{\left(5x^3\right)}^2-{\log }_a\sqrt{2x+3}\\ & =& {\log }_a\frac{{\left(5x^3\right)}^2}{\sqrt{2x+3}}\\ & =& {\log }_a\frac{25x^6}{\sqrt{2x+3}}\end{array}$

Step 3. Conclusion.

The obtained result is${\log }_a\frac{25x^6}{\sqrt{2x+3}}$.