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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. 20 (page 351)

Write $\log_{2} (\frac{4 x^{3}}{x^{2} - 3 x - 18})$ as the sum and/or difference of logarithms. Express powers as factors.

Short Answer

Expert verified

The expression can be written as $\begin{array}{rcl} 2 + 3 \log_{2} (x) - \log_{2} (x - 6) - \log_{2} (x + 3) \end{array}$

Step by step solution

01

Given Information

An expression is to be written in form of all logarithmic sum or differences.

$\log_{2} (\frac{4 x^{3}}{x^{2} - 3 x - 18})$

02

Properties of log

$\log (a b) = \log a + \log b$
$\log (\frac{a}{b}) = \log a - \log b$
$\log a^{n} = n \log a$
$\log_{a} (a^{n}) = n$

03

Simplifying

Given expression is $P = \log_{2} (\frac{4 x^{3}}{x^{2} - 3 x - 18})$

Using log properties we get,

$\begin{array}{rcl} P & = & \log_{2} (4 x^{3}) - \log_{2} (x^{2} - 3 x - 18) \\ P & = & \log_{2} (4) + \log_{2} (x^{3}) - \log_{2} [(x - 6) (x + 3)] \\ P & = & 2 + 3 \log_{2} (x) - \log_{2} (x - 6) - \log_{2} (x + 3) \end{array}$

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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.

True or False

To graph ${(x - 2)}^{3}$ , shift the graph of $y = x^{3}$ to the left $2$ units.

If $g (- 5) = 3$ and g is one-to-one, what is $g^{- 1} (3)$ ?

A function $f$ has an inverse function. If the graph of $f$ lies in quadrant II, in which quadrant does the graph of $f^{- 1}$ lie?

Evaluate each expression using the graphs of $y = f (x)$ and $y = g (x)$ shown in the figure.

(a) $(g 鈭� f) (- 1)$ (b) $(g 鈭� f) (0)$ (c) $(f 鈭� g) (- 1)$ (d) $(f 鈭� g) (4)$ .

As the base a of an exponential function $f (x) = a^{x}$ , where $a > 1$ increases, what happens to the behavior of its graph for $x > o ?$ What happens to the behavior of its graph for $x < 0 ?$

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