Chapter 10: Q. 10.67 (page 1042)

Use the Properties of Logarithms to expand the logarithm $\log_{4} \sqrt[5]{\frac{x^{4}}{2 y^{3} z^{2}}}$ . Simplify, if possible

Short Answer

Expert verified

The expanded form of the logarithm is $\log_{4} \sqrt[5]{\frac{x^{4}}{2 y^{3} z^{2}}} = \frac{1}{5} (4 \log_{4} x - \frac{1}{2} - 3 \log_{4} y - 2 \log_{4} z)$

Step by step solution

Step 1. given data

Given logarithm expressions is $\log_{4} \sqrt[5]{\frac{x^{4}}{2 y^{3} z^{2}}}$

Step 2. Apply power Property of the logarithm

Rewrite the radical with a rational exponent

$\log_{4} \sqrt[5]{\frac{x^{4}}{2 y^{3} z^{2}}} = \log_{4} {(\frac{x^{4}}{2 y^{3} z^{2}})}^{\frac{1}{5}}$

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

$\log_{4} \sqrt[5]{\frac{x^{4}}{2 y^{3} z^{2}}} = \log_{4} {(\frac{x^{4}}{2 y^{3} z^{2}})}^{\frac{1}{5}} \log_{4} \sqrt[5]{\frac{x^{4}}{2 y^{3} z^{2}}} = \frac{1}{5} \log_{4} (\frac{x^{4}}{2 y^{3} z^{2}})$

Step 3. Apply the Quotient Property of logarithm

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

solocalid="1644958854832" $\log_{4} \sqrt[5]{\frac{x^{4}}{2 y^{3} z^{2}}} = \frac{1}{5} \log_{4} (\frac{x^{4}}{2 y^{3} z^{2}}) \log_{4} \sqrt[5]{\frac{x^{4}}{2 y^{3} z^{2}}} = \frac{1}{5} (\log_{4} x^{4} - \log_{4} (2 y^{3} z^{2}))$

Step 4. apply product Properties of Logarithms

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

localid="1644959168978" $\log_{4} \sqrt[5]{\frac{x^{4}}{2 y^{3} z^{2}}} = \frac{1}{5} (\log_{4} x^{4} - \log_{4} (2 y^{3} z^{2})) \log_{4} \sqrt[5]{\frac{x^{4}}{2 y^{3} z^{2}}} = \frac{1}{5} (\log_{4} x^{4} - (\log_{4} 2 + \log_{4} y^{3} + \log_{4} z^{2})) \log_{4} \sqrt[5]{\frac{x^{4}}{2 y^{3} z^{2}}} = \frac{1}{5} (\log_{4} x^{4} - \log_{4} 2 - \log_{4} y^{3} - \log_{4} z^{2})$

Step 5. Apply power Property of the logarithm

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

$\log_{4} \sqrt[5]{\frac{x^{4}}{2 y^{3} z^{2}}} = \frac{1}{5} (\log_{4} x^{4} - \log_{4} 2 - \log_{4} y^{3} - \log_{4} z^{2}) \log_{4} \sqrt[5]{\frac{x^{4}}{2 y^{3} z^{2}}} = \frac{1}{5} (4 \log_{4} x - \frac{1}{2} - 3 \log_{4} y - 2 \log_{4} z)$

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Use the Properties of Logarithms to expand the logarithm $\log_{4} \sqrt[5]{\frac{x^{4}}{2 y^{3} z^{2}}}$ . 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

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Use the Properties of Logarithms to expand the logarithm log4x42y3z25. 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

Decision Maths

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Pure Maths

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Calculus

Study anywhere. Anytime. Across all devices.

Use the Properties of Logarithms to expand the logarithm $\log_{4} \sqrt[5]{\frac{x^{4}}{2 y^{3} z^{2}}}$ . Simplify, if possible