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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 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

01

Step 1. Given information.

Consider the given information.

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

02

Step 2. Formula used.

The logarithm properties are defined as,

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

03

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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Most popular questions from this chapter

A function $y = f (x)$ is decreasing on the interval $(0, 5) .$ What conclusions can you draw about the graph of $y = f^{- 1} (x)$ ?

Use the graph of $y = f (x)$ given in Problem $44$ to evaluate

the following:

(a) $f (2)$

(b) $f (1)$

(c) $f^{- 1} (0)$

(d) $f^{- 1} (- 1)$

solve each equation. Verify your results using a graphing utility.

$4^{x^{2}} = 2^{x}$

In Problems $33 - 40$ , the graph of an exponential function is given. Match each graph to one of the following functions.

Solve the equation $5^{x} = 5^{- 6}$

and verify your results using a graphing utility

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