Q. 424 In the following exercises, use ... [FREE SOLUTION]

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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. 424 (page 1065)

In the following exercises, use properties of logarithms to write each logarithm as a sum of logarithms. Simplify if possible.
$\ln \frac{\sqrt{3 x^{2} - y^{2}}}{z^{4}}$

Short Answer

Expert verified

logarithm expression as a sum of logarithms will be $\ln \frac{\sqrt{3 x^{2} - y^{2}}}{z^{4}} = \frac{1}{2} \ln 3 + \ln x - \ln y - 4 \ln z$

Step by step solution

Step 1. Given data

The given logarithm expression is $\ln \frac{\sqrt{3 x^{2} - y^{2}}}{z^{4}}$

Step 2. Apply the quotient properties of the logarithm

Use quotient properties of the logarithm $\ln (\frac{m}{n}) = \ln (m) - \ln (n)$

$\ln \frac{\sqrt{3 x^{2} - y^{2}}}{z^{4}} = \ln \sqrt{3 x^{2} - y^{2}} - \ln z^{4} \ln \frac{\sqrt{3 x^{2} - y^{2}}}{z^{4}} = \ln {(3 x^{2} - y^{2})}^{\frac{1}{2}} - \ln z^{4}$

Step 3. Apply the power properties of the logarithm

Use power properties of the logarithm $\ln (m^{p}) = p \ln (m)$

$\ln \frac{\sqrt{3 x^{2} - y^{2}}}{z^{4}} = \ln {(3 x^{2} - y^{2})}^{\frac{1}{2}} - \ln z^{4} \ln \frac{\sqrt{3 x^{2} - y^{2}}}{z^{4}} = \frac{1}{2} \ln (3 x^{2} - y^{2}) - 4 \ln z$

Step 4. Apply the quotient properties of the logarithm

Use quotient properties of the logarithm $\ln (\frac{m}{n}) = \ln (m) - \ln (n)$

role="math" localid="1645023256402" $\ln \frac{\sqrt{3 x^{2} - y^{2}}}{z^{4}} = \frac{1}{2} \ln (3 x^{2} - y^{2}) - 4 \ln z \ln \frac{\sqrt{3 x^{2} - y^{2}}}{z^{4}} = \frac{1}{2} \ln 3 x^{2} - \frac{1}{2} \ln y^{2} - 4 \ln z$

Step 5. Apply the product properties of logarithm

Use product property of logarithm $\ln (m 路 n) = \ln (m) + \ln (n)$

role="math" localid="1645023403775" $\ln \frac{\sqrt{3 x^{2} - y^{2}}}{z^{4}} = \frac{1}{2} \ln 3 x^{2} - \frac{1}{2} \ln y^{2} - 4 \ln z \ln \frac{\sqrt{3 x^{2} - y^{2}}}{z^{4}} = \frac{1}{2} \ln 3 + \frac{1}{2} \ln x^{2} - \frac{1}{2} \ln y^{2} - 4 \ln z$

Step 6. Apply the power properties of the logarithm

Use power properties of the logarithm $\ln (m^{p}) = p \ln (m)$

role="math" localid="1645023480021" $\ln \frac{\sqrt{3 x^{2} - y^{2}}}{z^{4}} = \frac{1}{2} \ln 3 + \frac{1}{2} \ln x^{2} - \frac{1}{2} \ln y^{2} - 4 \ln z \ln \frac{\sqrt{3 x^{2} - y^{2}}}{z^{4}} = \frac{1}{2} \ln 3 + \frac{2}{2} \ln x - \frac{2}{2} \ln y - 4 \ln z \ln \frac{\sqrt{3 x^{2} - y^{2}}}{z^{4}} = \frac{1}{2} \ln 3 + \ln x - \ln y - 4 \ln z$

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91影视

In the following exercises, use properties of logarithms to write each logarithm as a sum of logarithms. Simplify if possible.
$\ln \frac{\sqrt{3 x^{2} - y^{2}}}{z^{4}}$

Short Answer

Step by step solution

Step 1. Given data

Step 2. Apply the quotient properties of the logarithm

Step 3. Apply the power properties of the logarithm

Step 4. Apply the quotient properties of the logarithm

Step 5. Apply the product properties of logarithm

Step 6. Apply the power properties of the logarithm

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91影视

In the following exercises, use properties of logarithms to write each logarithm as a sum of logarithms. Simplify if possible.ln3x2-y2z4

Short Answer

Step by step solution

Step 1. Given data

Step 2. Apply the quotient properties of the logarithm

Step 3. Apply the power properties of the logarithm

Step 4. Apply the quotient properties of the logarithm

Step 5. Apply the product properties of logarithm

Step 6. Apply the power properties of the logarithm

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Calculus

Applied Mathematics

Decision Maths

Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.

In the following exercises, use properties of logarithms to write each logarithm as a sum of logarithms. Simplify if possible.
$\ln \frac{\sqrt{3 x^{2} - y^{2}}}{z^{4}}$