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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 48. (page 308)

Write each expression as a sum and/or difference of logarithms. Express powers as factors.
$\ln (x \sqrt{1 + x^{2}})$

Short Answer

Expert verified

The result is $\ln x + \frac{1}{2} \ln (1 + x^{2})$ .

Step by step solution

01

Step 1. Given information.

Consider the given information.

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

02

Step 2. Formula used.

The logarithm properties are defined as,

$\begin{array}{l} \log_{a} (M N) = \log_{a} M + \log_{a} N \\ \log_{a} M^{r} = r \log_{a} M \end{array}$

03

Step 3. Calculation.

Apply the logarithm property to simplify.

$\begin{array}{rcl} \ln (x \sqrt{1 + x^{2}}) & = & \ln x + \ln \sqrt{1 + x^{2}} \\ = & \ln x + \ln {(1 + x^{2})}^{\frac{1}{2}} \\ = & \ln x + \frac{1}{2} \ln (1 + x^{2}) \end{array}$

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