/*! 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 35 Use properties of logarithms to ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 35

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log _{5} \sqrt[3]{\frac{x^{2} y}{25}}$$

Short Answer

Expert verified
The expanded form of \( \log_5 { \sqrt[3] { \frac {x^2 y}{25} } } \) is \( \frac {2}{3} \log_5 { x } + \frac {1}{3} \log_5 { y } - \frac {2}{3} \).

Step by step solution

01

Apply the Third Rule of Logarithms

The third rule of logarithms states that \( \log_b { a^m } = m \log_b { a }\). But before we can use it, notice that the inside of the log is a cube root, which can be rewritten with an exponent of \( \frac {1}{3} \). So \( \log_5 { \sqrt[3] { \frac {x^2 y}{25} } } \) can be rewritten as \( \frac {1}{3} \log_5 { \frac {x^2 y}{25} }\).
02

Apply the First Rule of Logarithms

The first rule of logarithms states that \( \log_b { a \cdot c } = \log_b { a } + \log_b { c } \) and \( \log_b { a/c } = \log_b { a } - \log_b { c }\), where \( b, a, c > 0 \) and \( b \neq 1 \). So, the above expression can be further broken down as \( \frac {1}{3} (\log_5 { x^2 } + \log_5 { y } - \log_5 { 25 }) \).
03

Apply the Third Rule of Logarithms Again

Apply the third rule of logarithms again to break down \( \log_5 { x^2 }\) as \( 2 \log_5 { x } \). Now the expression looks like this: \( \frac {1}{3} (2\log_5 { x } + \log_5 { y } - \log_5 { 25 }) \).
04

Simplify the Expression

The last part to simplify is \( \log_5 { 25 }\). Since \( 5^2 \) is 25, \( \log_5 { 25 } = 2 \). Thus, substituting this into the equation gives the final expanded form of the logarithm: \( \frac {2}{3} \log_5 { x } + \frac {1}{3} \log_5 { y } - \frac {2}{3} \).

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

U.S. soldiers fight Russian troops who have invaded New York City. Incoming missiles from Russian submarines and warships ravage the Manhattan skyline. It's just another scenario for the multi-billion-dollar video games Call of Duty, which have sold more than 100 million games since the franchise's birth in 2003 . The table shows the annual retail sales for Call of Duty video games from 2004 through 2010 . Create a scatter plot for the data. Based on the shape of the scatter plot, would a logarithmic function, an exponential function, or a linear function be the best choice for modeling the data? $$\begin{array}{l|c} \hline \text { Annual Retail Sales for } \text {Call of Duty Games} \\ \hline \text { Year } & \begin{array}{c} \text { Retail Sales } \\ \text { (millions of dollars) } \end{array} \\ \hline 2004 & 56 \\ \hline 2005 & 101 \\ \hline 2006 & 196 \\ \hline 2007 & 352 \\ \hline 2008 & 436 \\ \hline 2009 & 778 \\ \hline 2010 & 980 \\ \hline \end{array}$$

Write as a single term that does not contain a logarithm: $$e^{\ln 8 x^{5}-\ln 2 x^{2}}$$

Describe the product rule for logarithms and give an example.

Will help you prepare for the material covered in the next section. Simplify: \(16^{\frac{3}{2}}\)

What is an exponential equation?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Pure Maths

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.