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

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.

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

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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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

Step by step solution

Step 1. Given and explanation

Step 2. Simplifying the expression

Step 3. Simplifying

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In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible.log3x2327y4

Short Answer

Step by step solution

Step 1. Given and explanation

Step 2. Simplifying the expression

Step 3. Simplifying

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Decision Maths

Probability and Statistics

Discrete Mathematics

Calculus

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Study anywhere. Anytime. Across all devices.

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