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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 5: Q 67. (page 309)

Write each expression as a single logarithm.
$2 \log_{a} (5 x^{3}) - \frac{1}{2} \log_{a} (2 x + 3)$

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given information.

Consider the given information.

$2 \log_{a} (5 x^{3}) - \frac{1}{2} \log_{a} (2 x + 3)$

02

Step 2. Calculation.

Apply the logarithm properties to combine the terms.

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

03

Step 3. Conclusion.

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

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