Chapter 5: Q 53. (page 308)

Write each expression as a sum and/or difference of logarithms. Express powers as factors.
$\ln {[\frac{x^{2} - x - 2}{{(x + 4)}^{2}}]}^{\frac{1}{3}}$

Short Answer

Expert verified

The result is $\frac{1}{3} [\ln (x - 2) + \ln (x + 1) - 2 \ln (x + 4)]$ .

Step by step solution

Step 1. Given information.

Consider the given information.

$\ln {[\frac{x^{2} - x - 2}{{(x + 4)}^{2}}]}^{\frac{1}{3}}$

Step 2. Formula used.

The logarithm properties are defined as,

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

Step 3. Calculation.

Apply the logarithm property to simplify.

$\begin{array}{rcl} \ln {[\frac{x^{2} - x - 2}{{(x + 4)}^{2}}]}^{\frac{1}{3}} & = & \frac{1}{3} \ln [\frac{x^{2} - x - 2}{{(x + 4)}^{2}}] \\ = & \frac{1}{3} [\ln (x^{2} - x - 2) - \ln {(x + 4)}^{2}] \\ = & \frac{1}{3} [\ln (x - 2) (x + 1) - 2 \ln (x + 4)] \\ = & \frac{1}{3} [\ln (x - 2) + \ln (x + 1) - 2 \ln (x + 4)] \end{array}$

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Write each expression as a sum and/or difference of logarithms. Express powers as factors.
$\ln {[\frac{x^{2} - x - 2}{{(x + 4)}^{2}}]}^{\frac{1}{3}}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Formula used.

Step 3. Calculation.

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Write each expression as a sum and/or difference of logarithms. Express powers as factors.lnx2-x-2x+4213

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Formula used.

Step 3. Calculation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Probability and Statistics

Theoretical and Mathematical Physics

Calculus

Statistics

Study anywhere. Anytime. Across all devices.

Write each expression as a sum and/or difference of logarithms. Express powers as factors.
$\ln {[\frac{x^{2} - x - 2}{{(x + 4)}^{2}}]}^{\frac{1}{3}}$