Chapter 10: Q. 260 (page 1045)

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

Short Answer

Expert verified

The simplified expression is,

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

Step by step solution

Step 1. Given and explanation

We are given an expression,

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

Use the properties of logarithms to simplify the given expression.

Step 2. Simplifying the expression

The expression can be written as,

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

Using power property of logarithm , we get

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

Using quotient property of logarithm , we get

role="math" localid="1645015725642" $\log_{2} \sqrt[4]{\frac{5 x^{3}}{2 y^{2} z^{4}}} = \frac{1}{4} (\log_{2} 5 x^{3} - \log_{2} 2 y^{2} z^{4}) \log_{2} \sqrt[4]{\frac{5 x^{3}}{2 y^{2} z^{4}}} = \frac{1}{4} \log_{2} 5 x^{3} - \frac{1}{4} \log_{2} 2 y^{2} z^{4}$

Step 3. Simplifying

Now, using product property of logarithm , we get

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

using power property of logarithm , we get

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

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.

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

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

Discrete Mathematics

Pure Maths

Applied Mathematics

Decision Maths

Geometry

Logic and Functions

Study anywhere. Anytime. Across all devices.

In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible.log25x32y2z44

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

Discrete Mathematics

Pure Maths

Applied Mathematics

Decision Maths

Geometry

Logic and Functions

Study anywhere. Anytime. Across all devices.

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