Chapter 10: Q. 467 (page 1068)

In the following exercises, use properties of logarithms to write each expression as a sum of logarithms, simplifying if possible
$\log_{2} \sqrt[4]{\frac{5 x^{3}}{16 y^{2} z^{7}}}$

Short Answer

Expert verified

The value is $\frac{1}{4} \log_{2} 5 + \frac{3}{4} \log_{2} x - 1 - \frac{1}{2} \log_{2} y - \frac{7}{4} \log_{2} z$

Step by step solution

Step 1. Given

An expression is given as $\log_{2} \sqrt[4]{\frac{5 x^{3}}{16 y^{2} z^{7}}}$

Step 2. Calculation

Rewrite the expression as,

$\log_{2} \sqrt[4]{\frac{5 x^{3}}{16 y^{2} z^{7}}} = \log_{2} {(\frac{5 x^{3}}{16 y^{2} z^{7}})}^{\frac{1}{4}} = \frac{1}{4} \log_{2} (\frac{5 x^{3}}{16 y^{2} z^{7}})$

Step 3. Further Calculation

Now use property $\log_{a} \frac{M}{N} = \log_{a} M - \log_{a} N$

$\frac{1}{4} \log_{2} (\frac{5 x^{3}}{2 y^{2} z^{4}}) = \frac{1}{4} (\log_{2} 5 x^{3} - \log_{2} 16 y^{2} z^{7}) = \frac{1}{4} \log_{2} 5 x^{3} - \frac{1}{4} \log_{2} 16 y^{2} z^{7}$

Now use the property $\log_{a} (M \cdot N) = \log_{a} M + \log_{a} N$

$\frac{1}{4} \log_{2} (\frac{5 x^{3}}{16 y^{2} z^{7}}) = \frac{1}{4} (\log_{2} 5 + \log_{2} x^{3}) - \frac{1}{4} (\log_{2} 16 + \log_{2} y^{2} + \log_{2} z^{7}) = \frac{1}{4} (\log_{2} 5 + 3 \log_{2} x) - \frac{1}{4} (\log_{2} 2^{4} + 2 \log_{2} y + 7 \log_{2} z) = \frac{1}{4} (\log_{2} 5 + 3 \log_{2} x) - \frac{1}{4} (4 + 2 \log_{2} y + 7 \log_{2} z) = \frac{1}{4} \log_{2} 5 + \frac{3}{4} \log_{2} x - 1 - \frac{1}{2} \log_{2} y - \frac{7}{4} \log_{2} z$

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In the following exercises, use properties of logarithms to write each expression as a sum of logarithms, simplifying if possible
$\log_{2} \sqrt[4]{\frac{5 x^{3}}{16 y^{2} z^{7}}}$

Short Answer

Step by step solution

Step 1. Given

Step 2. Calculation

Step 3. Further Calculation

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In the following exercises, use properties of logarithms to write each expression as a sum of logarithms, simplifying if possiblelog25x316y2z74

Short Answer

Step by step solution

Step 1. Given

Step 2. Calculation

Step 3. Further Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Pure Maths

Theoretical and Mathematical Physics

Calculus

Statistics

Study anywhere. Anytime. Across all devices.

In the following exercises, use properties of logarithms to write each expression as a sum of logarithms, simplifying if possible
$\log_{2} \sqrt[4]{\frac{5 x^{3}}{16 y^{2} z^{7}}}$