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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. 259 (page 1045)

In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible.
$\log_{3} \frac{\sqrt{3 x + 2 y^{2}}}{5 z^{2}}$

Short Answer

Expert verified

The simplified expression is,

$\frac{1}{2} \log_{3} (3 x + 2 y^{2}) - \log_{3} 5 + 2 \log_{3} z$

Step by step solution

01

Step 1. Given and explanation

We are given an expression,

$\log_{3} \frac{\sqrt{3 x + 2 y^{2}}}{5 z^{2}}$

Use the properties of logarithms to simplify the given expression.

02

Step 2. Simplifying the expression

Using quotient property of logarithm , we get

$\log_{3} \frac{\sqrt{3 x + 2 y^{2}}}{5 z^{2}} = \log_{3} \sqrt{3 x + 2 y^{2}} - \log_{3} 5 z^{2} \log_{3} \frac{\sqrt{3 x + 2 y^{2}}}{5 z^{2}} = \log_{3} {(3 x + 2 y^{2})}^{\frac{1}{2}} - \log_{3} 5 z^{2}$

03

Step 3. Simplifying

Now, using product property of logarithm , we get

$\log_{3} \frac{\sqrt{3 x + 2 y^{2}}}{5 z^{2}} = \log_{3} {(3 x + 2 y^{2})}^{\frac{1}{2}} - (\log_{3} 5 + \log_{3} z^{2})$

Using power property of logarithm , we get

localid="1645015082469" $\log_{3} \frac{\sqrt{3 x + 2 y^{2}}}{5 z^{2}} = \frac{1}{2} \log_{3} (3 x + 2 y^{2}) - \log_{3} 5 - 2 \log_{3} z$

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Most popular questions from this chapter

In the following exercises, graph each exponential function.

$f (x) = 2^{x}$

In the following exercises, find

a. $f 鈭� g (x)$

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c. $(f . g) (x)$

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$鈸� 2^{x} 鈸� 2^{x + 1} 鈸� 2^{x - 1} 鈸� 2^{x} + 2 鈸� 2^{x} 鈭� 2 鈸� 3^{x}$

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$y = \log_{7} (x)$

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