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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. 257 (page 1045)

In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible.
$\log_{3} \frac{\sqrt[3]{x^{2}}}{27 y^{4}}$

Short Answer

Expert verified

The simplified expression is,

$\frac{2}{3} \log_{3} x - 3 - 4 \log_{3} y$

Step by step solution

01

Step 1. Given and explanation

We are given an expression,

$\log_{3} \frac{\sqrt[3]{x^{2}}}{27 y^{4}}$

Use the properties of logarithms to simplify the given expression.

02

Step 2. Simplifying the expression

Using quotient property of logarithm , we get

$\log_{3} \frac{\sqrt[3]{x^{2}}}{27 y^{4}} = \log_{3} \sqrt[3]{x^{2}} - \log_{3} 27 y^{4}$

Using product property of logarithm , we get

role="math" localid="1645010871275" $\log_{3} \frac{\sqrt[3]{x^{2}}}{27 y^{4}} = \log_{3} x^{\frac{2}{3}} - (\log_{3} 27 + \log_{3} y^{4}) \log_{3} \frac{\sqrt[3]{x^{2}}}{27 y^{4}} = \log_{3} x^{\frac{2}{3}} - \log_{3} 27 - \log_{3} y^{4}$

03

Step 3. Simplifying

Simplifying,

$\log_{3} \frac{\sqrt[3]{x^{2}}}{27 y^{4}} = \log_{3} x^{\frac{2}{3}} - \log_{3} 3^{3} - \log_{3} y^{4}$

Now, using power property of logarithm , we get

$\log_{3} \frac{\sqrt[3]{x^{2}}}{27 y^{4}} = \frac{2}{3} \log_{3} x - 3 \log_{3} 3 - 4 \log_{3} y$

Using $\log_{a} a = 1$ , we get

$\log_{3} \frac{\sqrt[3]{x^{2}}}{27 y^{4}} = \frac{2}{3} \log_{3} x - 3 - 4 \log_{3} y$

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Most popular questions from this chapter

In the following exercises, solve each logarithmic equation

$y = \log_{\frac{1}{5}} (x)$

In the following exercises, graph each function in the same coordinate system:

$f (x) = 2^{x}, g (x) = 2^{x} - 1$

In the following exercise, find the inverse of each function.

$x^{3} - 4$

In the following exercise, find the inverse of each function.

$f (x) = \sqrt[3]{x + 5}$

In the following exercise, find the inverse of each function.

$f (x) = \sqrt[3]{x - 3}$

See all solutions

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