Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
![### Solving Exponential Equations
#### Problem Statement
Find the solution of the exponential equation:
\[ e^{2x + 1} = 46 \]
Express your answer in terms of logarithms, or correct to four decimal places.
#### Solution
First, we rewrite the equation:
\[ e^{2x + 1} = 46 \]
To isolate the exponent, we take the natural logarithm (ln) of both sides of the equation:
\[ \ln(e^{2x + 1}) = \ln(46) \]
Using the property of logarithms that \(\ln(e^y) = y\), we can simplify the left-hand side:
\[ 2x + 1 = \ln(46) \]
Next, we solve for \(x\):
\[ 2x + 1 = \ln(46) \]
\[ 2x = \ln(46) - 1 \]
\[ x = \frac{\ln(46) - 1}{2} \]
Thus, the solution in terms of logarithms is:
\[ x = \frac{\ln(46) - 1}{2} \]
#### Numerical Solution
To find the solution correct to four decimal places, we evaluate the logarithm and perform the arithmetic:
\[ \ln(46) \approx 3.8286 \]
Hence,
\[ x = \frac{3.8286 - 1}{2} = \frac{2.8286}{2} \approx 1.4143 \]
Therefore, the solution is:
\[ x \approx 1.4143 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb59be0f4-44c1-4a34-b2a8-1729261dae5c%2Fd79fef2b-0925-4f97-a429-d16e2927c3ff%2Fjocnxy8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Exponential Equations
#### Problem Statement
Find the solution of the exponential equation:
\[ e^{2x + 1} = 46 \]
Express your answer in terms of logarithms, or correct to four decimal places.
#### Solution
First, we rewrite the equation:
\[ e^{2x + 1} = 46 \]
To isolate the exponent, we take the natural logarithm (ln) of both sides of the equation:
\[ \ln(e^{2x + 1}) = \ln(46) \]
Using the property of logarithms that \(\ln(e^y) = y\), we can simplify the left-hand side:
\[ 2x + 1 = \ln(46) \]
Next, we solve for \(x\):
\[ 2x + 1 = \ln(46) \]
\[ 2x = \ln(46) - 1 \]
\[ x = \frac{\ln(46) - 1}{2} \]
Thus, the solution in terms of logarithms is:
\[ x = \frac{\ln(46) - 1}{2} \]
#### Numerical Solution
To find the solution correct to four decimal places, we evaluate the logarithm and perform the arithmetic:
\[ \ln(46) \approx 3.8286 \]
Hence,
\[ x = \frac{3.8286 - 1}{2} = \frac{2.8286}{2} \approx 1.4143 \]
Therefore, the solution is:
\[ x \approx 1.4143 \]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education