/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 66 Use the properties of logarithms... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 66

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $$\ln \sqrt{x^{2}(x+2)}$$

Short Answer

Expert verified
The expanded form of the expression \(\ln \sqrt{x^{2}(x+2)}\) is \(\frac{1}{2} \ln x + \frac{1}{4} \ln (x+2)\)

Step by step solution

01

Convert the square root into a fractional power

Any square root can be rewritten as a power of \(\frac{1}{2}\). So, the given expression becomes \(\ln (x^{2}(x+2))^\frac{1}{2}\)
02

Apply the power property of logarithm

The power rule for logarithms states that \(\ln a^b = b \ln a\) . Apply this rule: \(\ln (x^{2}(x+2))^{1/2} = \frac{1}{2} \ln (x^{2}(x+2))\)
03

Apply the product property of logarithm

The product property of logarithms states that \(\ln ab = \ln a + \ln b\). We apply this here: \(\frac{1}{2} \ln (x^{2}(x+2)) = \frac{1}{2} (\ln x^{2} + \ln (x+2))\)
04

Apply the power property of logarithm again

The power property allows us to bring the power of a logarithmic expression down, so we get \(\frac{1}{2} (\ln x^{2} + \ln (x+2)) = \ln x + \frac{1}{2} \ln (x+2)\)
05

Simplify the expression

Finally, distribute the 1/2 into the parantheses to get a simplified form: \(\frac{1}{2} \ln x + \frac{1}{4} \ln (x+2)\)

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Carbon 14 dating assumes that the carbon dioxide on Earth today has the same radioactive content as it did centuries ago. If this is true, the amount of \({ }^{14} \mathrm{C}\) absorbed by a tree that grew several centuries ago should be the same as the amount of \({ }^{14} \mathrm{C}\) absorbed by a tree growing today. A piece of ancient charcoal contains only \(15 \%\) as much radioactive carbon as a piece of modern charcoal. How long ago was the tree burned to make the ancient charcoal if the half-life of \({ }^{14} \mathrm{C}\) is 5715 years?

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log _{3} x+\log _{3}(x-8)=2$$

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$e^{-2 x}-2 x e^{-2 x}=0$$

Determine the principal \(P\) that must be invested at rate \(r\), compounded monthly, so that $$\$ 500,000$$ will be available for retirement in \(t\) years. $$r=3 \frac{1}{2} \%, t=15$$

The number of bacteria in a culture is increasing according to the law of exponential growth. The initial population is 250 bacteria, and the population after 10 hours is double the population after 1 hour. How many bacteria will there be after 6 hours?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.