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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.24 (page 349)

Write each expression as the sum and/or difference of logarithms. Express powers as factors.
$\log (x^{2} \sqrt{x^{3} + 1}), x > 0$

Short Answer

Expert verified

$\log (x^{2} \sqrt{x^{3} + 1}) = 2 \log x + \frac{1}{2} \log (x^{3} + 1)$

Step by step solution

01

Given data

The expression is: $\log (x^{2} \sqrt{x^{3} + 1}), x > 0$

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

$\log (x^{2} \sqrt{x^{3} + 1}) = \log x^{2} + \log \sqrt{x^{3} + 1}$ ; Using property of logarithms

$\log (x^{2} \sqrt{x^{3} + 1}) = \log x^{2} + \log {(x^{3} + 1)}^{\frac{1}{2}}$ ; Since $a^{\frac{1}{2}} = a^{- 1}$

role="math" localid="1646483035132" $\log (x^{2} \sqrt{x^{3} + 1}) = 2 \log x + \frac{1}{2} \log (x^{3} + 1)$ ; Using property of logarithms

04

Conclusion

Thus, $\log (x^{2} \sqrt{x^{3} + 1}) = 2 \log x + \frac{1}{2} \log (x^{3} + 1)$

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