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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.23 (page 349)

Write each expression as the sum and/or difference of logarithms. Express powers as factors.
$\log_{2} {(a^{2} \sqrt{b})}^{4}, a > 0, b > 0$

Short Answer

Expert verified

$\log_{2} {(a^{2} \sqrt{b})}^{4} = 8 \log_{2} a + 2 \log_{2} b$

Step by step solution

01

Given data

The expression is : $\log_{2} {(a^{2} \sqrt{b})}^{4}, a > 0, b > 0$

02

Concept used

Properties of logarithms:

$\log_{a} (M N) = \log_{a} M + \log_{a} N$
$\log_{a} (\frac{M}{N}) = \log_{a} M - \log_{a} N$
$\log_{a} M^{r} = r \log_{a} M$
$a^{x} = e^{x \ln a}$

03

Evaluate

$\log_{2} {(a^{2} \sqrt{b})}^{4} = 4 \log_{2} (a^{2} \sqrt{b})$ ; Using property of logarithms

$\log_{2} {(a^{2} \sqrt{b})}^{4} = 4 (\log_{2} a^{2} + \log_{2} \sqrt{b})$ ; Using property of logarithms

$\log_{2} {(a^{2} \sqrt{b})}^{4} = 4 \log_{2} a^{2} + 4 \log_{2} \sqrt{b}$ ; By distributive law

$\log_{2} {(a^{2} \sqrt{b})}^{4} = 4 \log_{2} a^{2} + 4 \log_{2} b^{\frac{1}{2}}$ ; Since $\sqrt{a} = a^{\frac{1}{2}}$

$\log_{2} {(a^{2} \sqrt{b})}^{4} = 4 路 2 \log_{2} a + 4 路 \frac{1}{2} \log_{2} b$ ; Using property of logarithms

$\log_{2} {(a^{2} \sqrt{b})}^{4} = 8 \log_{2} a + 2 \log_{2} b$ ; Simplifying

04

Conclusion

Thus, $\log_{2} {(a^{2} \sqrt{b})}^{4} = 8 \log_{2} a + 2 \log_{2} b$

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Most popular questions from this chapter

True or False. The domain of the composite function $(f 鈭� g) (x)$ is the same as the domain of $g (x)$ .

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.

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Height and Head Circumference. The head circumference

C of a child is related to the height H of the child (both in inches) through the function $H (C) = 2.15 C - 10.53$

(a) Express the head circumference C as a function of height H.

(b) Verify that $C = C (H)$ is the inverse of $H = H (C)$ by showing that $H (C (H)) = H$ and $C (H (C)) = C$ .

(c) Predict the head circumference of a child who is $26$ inches tall.

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