/*! 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 92 In Exercises 75-102, solve the l... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 92

In Exercises 75-102, solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$ \ln (x+5)=\ln (x-1)-\ln (x+1) $$

Short Answer

Expert verified
Based on the steps of the solution provided above, the solution of the given logarithmic equation is \(x=-1\).

Step by step solution

01

Apply the properties of logarithms

First, use the property of logarithms \(\ln a - \ln b = \ln(\frac{a}{b})\) to simplify the right side of the equation. So the equation \(\ln (x+5)=\ln (x-1) - \ln (x+1)\) becomes \(\ln (x+5)= \ln \left(\frac{x-1}{x+1}\right)\).
02

Remove the logarithm

Since both sides of the equality are the natural logarithms of two expressions, we can write the equation without the logarithm as \(x+5=\frac{x-1}{x+1}\).
03

Solve the resulting equation

Solving this new equation requires cross-multiplication and simplification: \(x + 5 = \frac{x-1}{x+1} \Rightarrow (x+5)(x+1)=(x-1) \Rightarrow x^2+6x+5=x-1 \Rightarrow x^2+7x+6=0\).
04

Find the roots

The resulting quadratic equation can be factored as \((x+1)(x+6)=0\), which yields the roots \(x=-1\) and \(x=-6\).
05

Check the roots in the domain

However, only \(x=-1\) is within the domain of the original logarithmic equation, so it is the only solution. Taking the anti-logarithm of a nonpositive number in the natural log function would result in an invalid operation (taking a log of a non-positive number). So we discard the root \(x=-6\).

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

\(e^{-0.5}=0.6065 \ldots\)

Students in a mathematics class were given an exam and then retested monthly with an equivalent exam. The average scores for the class are given by the human memory model \(f(t)=80-17 \log (t+1)\), \(0 \leq t \leq 12\) where \(t\) is the time in months. (a) Use a graphing utility to graph the model over the specified domain. (b) What was the average score on the original exam \((t=0) ?\) (c) What was the average score after 4 months? (d) What was the average score after 10 months?

\(g(x)=2 \ln x\) \(x=0.75\)

In Exercises 93-96, sketch the graph of \(f\) and \(g\) and describe the relationship between the graphs of \(f\) and \(g\). What is the relationship between the functions \(f\) and \(g\) ? \(f(x)=3^{x}, \quad g(x)=\log _{3} x\)

In Exercises 79-86, use the One-to-One Property to solve the equation for \(x\). \(\log _{2}(x+1)=\log _{2} 4\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Statistics

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.