/*! 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 26 Rewrite each expression as a sum... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 26

Rewrite each expression as a sum or difference of logarithms. $$\ln \left(\frac{a+b}{b}\right)$$

Short Answer

Expert verified
\(\ln (a+b) - \ln (b)\)

Step by step solution

01

Identify the Logarithm Property

Observe that the given logarithmic expression can be broken down using properties of logarithms. Specifically, the property \(\ln \left( \frac{M}{N} \right) = \ln M - \ln N\) is useful here.
02

Apply the Logarithm Property

Using the property \(\ln \left( \frac{M}{N} \right) = \ln M - \ln N\), rewrite the given expression \(\ln \left( \frac{a+b}{b} \right)\) as \(\ln (a+b) - \ln (b)\).
03

Simplify the Expression

After applying the logarithm property, the expression is now: \(\ln (a+b) - \ln (b)\). This is the desired form as a difference of logarithms.

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.

logarithmic expressions
Logarithmic expressions involve the logarithm function, which is the inverse of exponentiation. If you understand exponentiation, you'll quickly grasp logarithms too. A logarithm answers the question: to what power must we raise a base, usually 10 or the natural base e, to get a certain number?
For example, in the equation \(\text{log}_b (x) = y\), b is the base, x is the number, and y is the exponent. This implies that b raised to the power y gives you x (\(b^y = x\)).
Understanding logarithmic expressions is critical for solving equations involving exponential growth, decay, and other phenomena where relationships change multiplicatively.
logarithmic properties
Logarithmic properties make dealing with logarithms easier by converting multiplicative and divisive relationships into additive and subtractive ones. The main properties of logarithms are:
  • The Product Property: \(\text{log}_b (MN) = \text{log}_b (M) + \text{log}_b (N)\).
  • The Quotient Property: \(\text{log}_b \left(\frac{M}{N}\right) = \text{log}_b (M) - \text{log}_b (N)\).
  • The Power Property: \(\text{log}_b (M^p) = p \cdot \text{log}_b (M)\).

These properties help simplify complex logarithmic expressions, making it easier to solve various mathematical problems.
In the given exercise, we use the Quotient Property to rewrite the problem.
simplifying logarithms
Simplifying logarithms involves using logarithmic properties to make expressions as straightforward as possible. Start by identifying which properties of logarithms are applicable to your problem.
For instance: If you have \(\ln \left(\frac{a+b}{b}\right)\), recognize that the Quotient Property can be applied. Using this property, you break the expression into two simpler parts:
\(\ln(a+b) - \ln(b)\).
This makes the problem easier to understand and solve.
End with checking your final expression to ensure no further simplification is possible.
difference of logarithms
To convert a logarithmic fraction into a difference of logarithms, use the Quotient Property (\text{log}_b \(\left(\frac{M}{N}\right)\) = \text{log}_b \(M\) - \text{log}_b \(N\)).
In the exercise, \(\ln \left(\frac{a+b}{b}\right)\) is rewritten by separating the fraction's numerator and denominator into distinct logarithmic terms.
So, applying the Quotient Property translates to:
\(\ln (a+b) - \ln (b)\).
This reformatting clarifies the relationships within the original logarithmic expression and makes it simpler to manipulate or solve further.
Understanding how to convert between these forms is fundamental in both algebra and calculus.

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

Computers per Capita The number of personal computers per 1000 people in the United States from 1990 through 2010 is given in the accompanying table (Consumer Industry Almanac, www.c-i-a.com). a. Use exponential regression on a graphing calculator to find the best- fitting curve of the form \(y=a \cdot b^{x},\) where \(x=0\) corresponds to 1990. b. Write your equation in the form \(y=a e^{e x}.\) c. Assuming that the number of computers per 1000 people is growing continuously, what is the annual percentage rate? d. In what year will the number of computers per 1000 people reach \(1500 ?\) e. Judging from the graph of the data and the curve, does the exponential model look like a good model? $$\begin{array}{|l|c|} \hline \text { Year } & \begin{array}{c} \text { Computers } \\ \text { per 1000 } \end{array} \\ \hline 1990 & 192 \\ 1995 & 321 \\ 2000 & 628 \\ 2005 & 778 \\ 2010 & 932 \\ \hline \end{array}$$

Find the approximate solution to each equation. Round to four decimal places. $$5 e^{x}=4$$

Time of Death A detective discovered a body in a vacant lot at 7 A.M. and found that the body temperature was \(80^{\circ} \mathrm{F}\). The county coroner examined the body at 8 A.M. and found that the body temperature was \(72^{\circ} .\) Assuming that the body temperature was \(98^{\circ}\) when the person died and that the air temperature was a constant \(40^{\circ}\) all night, what was the approximate time of death?

Factor \(3 x-9+w x-3 w\)

Present Value of a \(\mathrm{CD}\) What amount (present value) must be deposited today in a certificate of deposit so that the investment will grow to \(\$ 20,000\) in 18 years at \(6 \%\) compounded continuously.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

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.