/*! 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 88 Solve each logarithmic equation.... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 88

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log (x+3)+\log (x-2)=\log 14$$

Short Answer

Expert verified
The solution to the given equation is \(x = 4.00\).

Step by step solution

01

Combine the Logarithmic Expressions

Based on the properties of logarithms, we know that the logarithm of a product is the sum of the logarithms. Therefore, the equation can be rewritten as: \(\log [(x+3)(x-2)] = \log 14\).
02

Remove the Logarithms

The equation has now taken the form \(\log_b m = \log_b n\), where \(b\), \(m\), and \(n\) are positive numbers and \(b \neq 1\). According to the equality property of logarithms, if two logarithms with the same base are equal, then their arguments are also equal. Therefore, we can remove the logarithms from our equation to get \((x+3)(x-2) = 14\).
03

Solve for \(x\)

We now have a quadratic equation. First, we expand it: \(x^2 + x - 6 = 14\). This simplifies to \(x^2 + x - 20 = 0\). By factoring, we get \((x-4)(x+5) = 0\). Setting each factor equal to zero gives us the potential solutions \(x = 4\) and \(x = -5\).
04

Check the Domain

We need to check if our solutions don't make the original logarithm expressions undefined. Testing \(x = 4\) in the original equation, we have \(\log (4+3) + \log (4-2)\) which gives \(\log 7 + \log 2 = \log 14\), as required. The value \(x = -5\) makes the term '\(x-2\)' negative, which is undefined for real number logarithms. Therefore, it must be rejected.
05

Approximate the Solution

The only solution which is in the domain of the original logarithmic expressions is \(x = 4\). In decimal form, this is already correct to two decimal places.

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

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because the equations $$\log (3 x+1)=5 \text { and } \log (3 x+1)=\log 5$$ are similar, I solved them using the same method.

Students in a mathematics class took a final examination. They took equivalent forms of the exam in monthly intervals thereafter. The average score, \(f(t),\) for the group after \(t\) months was modeled by the human memory function \(f(t)=75-10 \log (t+1), \quad\) where \(\quad 0 \leq t \leq 12 . \quad\) Use \(\quad\) a graphing utility to graph the function. Then determine how many months elapsed before the average score fell below 65.

Hurricanes are one of nature's most destructive forces. These low-pressure areas often have diameters of over 500 miles. The function \(f(x)=0.48 \ln (x+1)+27\) models the barometric air pressure, \(f(x),\) in inches of mercury, at a distance of \(x\) miles from the eye of a hurricane. The function \(P(t)=145 e^{-0.092 t}\) models a runner's pulse, \(P(t),\) in beats per minute, \(t\) minutes after a race, where \(0 \leq t \leq 15 .\) Graph the function using a graphing utility. TRACE along the graph and determine after how many minutes the runner's pulse will be 70 beats per minute. Round to the nearest tenth of a minute. Verify your observation algebraically.

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution=set. Verify this value by direct substitution into the equation. $$3^{x}=2 x+3$$

Use the formula \(t=\frac{\ln 2}{k}\) that gives the time for a population with a growth rate \(k\) to double to solve Exercises \(35-36 .\) Express each answer to the nearest whole year. The logistic growth function $$ f(t)=\frac{100,000}{1+5000 e^{-t}} $$ describes the number of people, \(f(t),\) who have become ill with influenza \(t\) weeks after its initial outbreak in a particular community. a. How many people became ill with the flu when the epidemic began? b. How many people were ill by the end of the fourth week? c. What is the limiting size of the population that becomes ill?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

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.