/*! 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 1 Write each expression in terms o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 1

Write each expression in terms of individual logarithms of \(x, y,\) and \(z\). a) \(\log _{7} x y^{2} \sqrt{z}\) b) \(\log _{5}(x y z)^{8}\) c) \(\log \frac{x^{2}}{y \sqrt[3]{z}}\) d) \(\log _{3} x \sqrt{\frac{y}{z}}\)

Short Answer

Expert verified
a) \(\log _{7} x + 2\log _{7} y + \frac{1}{2} \log _{7} z\) b) \(8(\log _{5} x + \log _{5} y + \log _{5} z)\) c) \(2 \log x - \log y - \frac{1}{3} \log z\) d) \(\log _{3} x + \frac{1}{2} \log _{3} y - \frac{1}{2} \log _{3} z\)

Step by step solution

01

Title - Expand logarithm using product rule

For expression a) \[\log _{7} x y^{2} \sqrt{z} = \log _{7} (x) + \log _{7} (y^{2}) + \log _{7} (\sqrt{z})\]
02

Title - Apply power rule to each term

Using the power rule of logarithms, \[\log _{7} x y^{2} \sqrt{z} = \log _{7} x + 2\log _{7} y + \frac{1}{2} \log _{7} z\]
03

Title - Expand using power and product rules

For expression b) \[\log _{5}(x y z)^{8} = 8 \log _{5}(x y z)\]
04

Title - Apply product rule inside the logarithm

Using the product rule of logarithms, \[8 \log _{5}(x y z) = 8(\log _{5}(x) + \log _{5}(y) + \log _{5}(z))\]
05

Title - Expand using quotient and power rules

For expression c) \[\log \frac{x^{2}}{y \sqrt[3]{z}} = \log (x^{2}) - \log (y \sqrt[3]{z})\]
06

Title - Apply product rule inside the logarithm

Using the product rule of logarithms, \[\log (y \sqrt[3]{z}) = \log (y) + \log (\sqrt[3]{z})\]
07

Title - Apply power rule to each term

Using the power rule of logarithms, \[\log x^{2} - (\log y + \frac{1}{3} \log z) = 2 \log x - \log y - \frac{1}{3} \log z\]
08

Title - Expand using quotient and power rules

For expression d) \[\log _{3} x \sqrt{\frac{y}{z}} = \log _{3} x + \log _{3} \sqrt{\frac{y}{z}}\]
09

Title - Apply the power and quotient rules

Using the power rule and applying them to the roots, \[= \log _{3} x + \frac{1}{2} \log _{3} \frac{y}{z} = \log _{3} x + \frac{1}{2}(\log _{3} y - \log _{3} z) = \log _{3} x + \frac{1}{2} \log _{3} y - \frac{1}{2} \log _{3} z\]

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.

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

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithm Properties
Understanding logarithm properties is essential for expanding and simplifying logarithmic expressions. These properties help you manipulate logarithms to make calculations simpler.

Here are the main properties to remember:
  • The **Product Rule**: \(\text{log}_b (MN) = \text{log}_b (M) + \text{log}_b (N)\)
  • The **Quotient Rule**: \(\text{log}_b \frac{M}{N} = \text{log}_b (M) - \text{log}_b (N)\)
  • The **Power Rule**: \(\text{log}_b (M^k) = k \text{log}_b (M)\)
Each of these rules makes it easier to break down complex logarithmic expressions into simpler terms. These properties act as tools that let you rewrite logarithms in a more manageable form.
Product Rule in Logarithms
The product rule in logarithms allows us to split the logarithm of a product into a sum of individual logarithms.

Example:
Consider \(\text{log}_7 (xy^2\text{鈭殅z)\). Applying the product rule, we get:

\(\text{log}_7 (xy^2\text{鈭殅z) = \text{log}_7 (x) + \text{log}_7 (y^2) + \text{log}_7 (\text{鈭殅z)\)

This makes it easier to apply further rules like the power rule.

Breaking down an expression using the product rule helps to simplify calculations.
Power Rule in Logarithms
The power rule in logarithms is used to move an exponent out in front of the logarithm. This can be very useful for simplifying logarithmic expressions.

Example:
For \(\text{log}_7 (y^2)\), we can apply the power rule:

\(\text{log}_7 (y^2) = 2 \text{log}_7 (y)\)

Similarly:

\(\text{log}_7 (\text{鈭殅z) = \text{log}_7 (z^{1/2}) = \frac{1}{2}\text{log}_7 (z)\)

Applying the power rule makes it easier to interact with the terms individually.
Quotient Rule in Logarithms
The quotient rule lets you split the logarithm of a fraction into the difference between the logarithms of the numerator and the denominator.

Example:
Consider \(\text{log} \frac{x^2}{y \text{鈭毬硙z}\). Using the quotient rule, we get:

\(\text{log} \frac{x^2}{y \text{鈭毬硙z} = \text{log} (x^2) - \text{log} (y \text{鈭毬硙z)\)

We can then apply the product rule and power rule within these terms for further simplification:

\(\text{log} (x^2) - (\text{log} (y) + \text{log} (\text{鈭毬硙z)) = 2\text{log}(x) - \text{log}(y) - \frac{1}{3} \text{log}(z)\)

This process will help in breaking down the complex fraction into simpler parts easier to work with.

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

If \(\log _{3} m=n,\) then determine \(\log _{3} m^{4},\) in terms of \(n\).

The graph of \(f(x)=\log _{2} x\) has been transformed to \(g(x)=a \log _{2} x+k .\) The transformed image passes through the points \(\left(\frac{1}{4},-9\right)\) and \((16,-6) .\) Determine the values of \(a\) and \(k\).

The growth of a new social networking site can be modelled by the exponential function \(N(t)=1.1^{t},\) where \(N\) is the number of users after \(t\) days. a) Write the equation of the inverse. b) How long will it take, to the nearest day, for the number of users to exceed \(1 000 000 ?\)

The compound interest formula is \(A=P(1+i)^{n},\) where \(A\) is the future amount, \(P\) is the present amount or principal, \(i\) is the interest rate per compounding period expressed as a decimal, and \(n\) is the number of compounding periods. All interest rates are annual percentage rates (APR). a) David inherits \(\$ 10\) 000 and invests in a guaranteed investment certificate (GIC) that earns \(6 \%,\) compounded semi-annually. How long will it take for the GIC to be worth \$11 000? b) Linda used a credit card to purchase a S1200 laptop computer. The rate of interest charged on the overdue balance is \(28 \%\) per year, compounded daily. How many days is Linda's payment overdue if the amount shown on her credit card statement is \(\$ 1241.18 ?\) c) How long will it take for money invested at \(5.5 \%,\) compounded semi- annually, to triple in value?

The formula for the Richter magnitude, \(M\) of an earthquake is \(M=\log \frac{A}{A_{0}},\) where \(A\) is the amplitude of the ground motion and \(A_{0}\) is the amplitude of a standard earthquake. In \(1985,\) an earthquake with magnitude 6.9 on the Richter scale was recorded in the Nahanni region of the Northwest Territories. The largest recorded earthquake in Saskatchewan occurred in 1982 near the town of Big Beaver. It had a magnitude of 3.9 on the Richter scale. How many times as great as the seismic shaking of the Saskatchewan earthquake was that of the Nahanni earthquake?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

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.