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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 54. (page 308)

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

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given Information.

Consider the given information.

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

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.

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

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Most popular questions from this chapter

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.

$A (n)$ ________ is a function of the form $f (x) = C a^{x}$ where $a > 0, a 鈮� 1,$ and $C 鈮� 0$ are real numbers. The base $a$ is a ____ and $C$ is a _____.

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

$e^{x} = e^{3 x + 8}$

Begin with the graph of $y = e^{x}$ and use transformation to graph the function. Determine the domain, range and horizontal asymptote of the function.

$f (x) = - e^{x}$

If $5^{- x} = 3$ , what does $5^{3 x}$ equal?

Solve the equation ${(\frac{1}{4})}^{x} = \frac{1}{64}$

and verify your results using a graphing utility.

See all solutions

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