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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 10: Q. 429 (page 1065)

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.
$\frac{1}{4} \log y - 2 \log (y - 3)$

Short Answer

Expert verified

a condensed form of the logarithm is $\frac{1}{4} \log y - 2 \log (y - 3) = \log \frac{\sqrt[4]{y}}{{(y - 3)}^{2}}$

Step by step solution

01

Step 1. Given data

The given logarithm expression is $\frac{1}{4} \log y - 2 \log (y - 3)$

02

Step 2. Apply the power properties of the logarithm

Use power properties of the logarithm $\log_{a} (m^{p}) = p \log_{a} (m)$

role="math" localid="1645031594030" $\frac{1}{4} \log y - 2 \log (y - 3) = \log y^{\frac{1}{4}} - \log {(y - 3)}^{2} \frac{1}{4} \log y - 2 \log (y - 3) = \log \sqrt[4]{y} - \log {(y - 3)}^{2}$

03

Step 3. Apply the quotient properties of the logarithm

Use quotient properties of the logarithm $\log (m) - \log (n) = \log (\frac{m}{n})$

role="math" localid="1645031603303" $\frac{1}{4} \log y - 2 \log (y - 3) = \log \sqrt[4]{y} - \log {(y - 3)}^{2} \frac{1}{4} \log y - 2 \log (y - 3) = \log \frac{\sqrt[4]{y}}{{(y - 3)}^{2}}$

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