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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 50. (page 308)

Write each expression as a sum and/or difference of logarithms. Express powers as factors.
$\log_{5} (\frac{\sqrt[3]{x^{2} + 1}}{x^{2} - 1})$

Short Answer

Expert verified

The result is $\begin{array}{rcl} \frac{1}{3} \log_{5} (x^{2} + 1) + \log_{5} (x + 1) + \log_{5} (x - 1) \end{array}$ .

Step by step solution

01

Step 1. Given information.

Consider the given information.

$\log_{5} (\frac{\sqrt[3]{x^{2} + 1}}{x^{2} - 1})$

02

Step 2. Formula used.

The logarithm properties are defined as,

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

03

Step 3. Calculation.

Apply the logarithm property to simplify.

$\begin{array}{rcl} \log_{5} (\frac{\sqrt[3]{x^{2} + 1}}{x^{2} - 1}) & = & \log_{5} (\sqrt[3]{x^{2} + 1}) + \log_{5} (x^{2} - 1) \\ = & \log_{5} {(x^{2} + 1)}^{\frac{1}{3}} + \log_{5} (x + 1) (x - 1) \\ = & \frac{1}{3} \log_{5} (x^{2} + 1) + \log_{5} (x + 1) + \log_{5} (x - 1) \end{array}$

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Most popular questions from this chapter

Vehicle Stopping Distance Taking into account reaction time, the distance $d$ (in feet) that a car requires to come to a complete stop while traveling $r$ miles per hour is given by the function

localid="1646198784356" $d (r) = 6.97 r - 90.39$

a. Express the speed $r$ at which the car is traveling as a function of the distance localid="1646198792401" $d$ required to come to a complete stop

b. Verify that localid="1646198797631" $r = r (d)$ is the inverse of localid="1646198802104" $d = d (r)$ by showing that localid="1646198806352" $r (d (r)) = r$ and localid="1646198811055" $d (r (d)) = d$

c. Predict the speed that a car was traveling if the distance required to stop was localid="1646198818232" $300$ feet.

$d (r (d)) = d$

The domain of one-to-one function g is $[0, 15]$ , and range is $(0, 8)$ . State the domain and range oflocalid="1646180141229" $g^{- 1}$

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.

$f (x) = 9 - 3 e^{- x}$

Determine the exponential function whose graph is given.

If $f (x) = a^{x}$ , show that $\frac{f (x + h) - f (x)}{h} = a^{x} . \frac{a^{h} - 1}{h} h 鈮� 0$ .

See all solutions

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