Chapter 10: Q. 10.68 (page 1042)

Use the Properties of Logarithms to expand the logarithm $\log_{3} \sqrt[3]{\frac{x^{2}}{5 y z}}$ . Simplify, if possible.

Short Answer

Expert verified

The expanded form of the logarithm is $\log_{3} \sqrt[3]{\frac{x^{2}}{5 y z}} = \frac{1}{3} (2 \log_{3} x - \log_{3} 5 - \log_{3} y - \log_{3} z)$

Step by step solution

Step 1. given data

Given logarithm expressions is $\log_{3} \sqrt[3]{\frac{x^{2}}{5 y z}}$

Step 2. Apply power Property of the logarithm

Rewrite the radical with a rational exponent

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

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

localid="1644959659341" $\log_{3} \sqrt[3]{\frac{x^{2}}{5 y z}} = \log_{3} {(\frac{x^{2}}{5 y z})}^{\frac{1}{3}} \log_{3} \sqrt[3]{\frac{x^{2}}{5 y z}} = \frac{1}{3} \log_{3} (\frac{x^{2}}{5 y z})$

Step 3. Apply the Quotient Property of logarithm

Use the Quotient Property of logarithm $\log_{a} (\frac{m}{n}) = \log_{a} (m) - \log_{a} (n)$

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

Step 4. apply product Properties of Logarithms

Use product Properties of Logarithm $\log_{a} (m \cdot n) = \log_{a} m + \log_{a} n$

localid="1644959953874" $\log_{3} \sqrt[3]{\frac{x^{2}}{5 y z}} = \frac{1}{3} (\log_{3} x^{2} - \log_{3} 5 y z) \log_{3} \sqrt[3]{\frac{x^{2}}{5 y z}} = \frac{1}{3} (\log_{3} x^{2} - (\log_{3} 5 + \log_{3} y + \log_{3} z)) \log_{3} \sqrt[3]{\frac{x^{2}}{5 y z}} = \frac{1}{3} (\log_{3} x^{2} - \log_{3} 5 - \log_{3} y - \log_{3} z)$

Step 5. Apply power Property of the logarithm

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

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

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Use the Properties of Logarithms to expand the logarithm $\log_{3} \sqrt[3]{\frac{x^{2}}{5 y z}}$ . Simplify, if possible.

Short Answer

Step by step solution

Step 1. given data

Step 2. Apply power Property of the logarithm

Step 3. Apply the Quotient Property of logarithm

Step 4. apply product Properties of Logarithms

Step 5. Apply power Property of the logarithm

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Mechanics Maths

Decision Maths

Logic and Functions

Discrete Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Use the Properties of Logarithms to expand the logarithm log3x25yz3. Simplify, if possible.

Short Answer

Step by step solution

Step 1. given data

Step 2. Apply power Property of the logarithm

Step 3. Apply the Quotient Property of logarithm

Step 4. apply product Properties of Logarithms

Step 5. Apply power Property of the logarithm

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Mechanics Maths

Decision Maths

Logic and Functions

Discrete Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Use the Properties of Logarithms to expand the logarithm $\log_{3} \sqrt[3]{\frac{x^{2}}{5 y z}}$ . Simplify, if possible.