Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Concept explainers
Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
![### Solving the Exponential Equation
To solve the exponential equation for \( x \):
\[ e^{5x} = 2 \]
Follow these steps:
1. **Take the natural logarithm on both sides.**
\[ \ln(e^{5x}) = \ln(2) \]
2. **Use the property of logarithms to simplify.**
\[ 5x \cdot \ln(e) = \ln(2) \]
Since \( \ln(e) = 1 \), the equation simplifies to:
\[ 5x = \ln(2) \]
3. **Solve for \( x \).**
\[ x = \frac{\ln(2)}{5} \]
Thus, the exact solution for \( x \) is:
\[ x = \frac{\ln(2)}{5} \]
(Type your answer in the provided blue box as shown in the equation.)
**Diagram Explanation:**
There is no graph or diagram associated with this problem. The equation involves exponential and logarithmic properties which are transformed for solving \( x \).
**Note:**
Be sure to type the exact answer \( \frac{\ln(2)}{5} \) in decimal or fraction form as required by your assignment format.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd8c403c1-73da-4878-97dd-7583351688c0%2Ff62a7b43-713c-43ed-bf29-dd4d5e271e8a%2Fl994h9j_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving the Exponential Equation
To solve the exponential equation for \( x \):
\[ e^{5x} = 2 \]
Follow these steps:
1. **Take the natural logarithm on both sides.**
\[ \ln(e^{5x}) = \ln(2) \]
2. **Use the property of logarithms to simplify.**
\[ 5x \cdot \ln(e) = \ln(2) \]
Since \( \ln(e) = 1 \), the equation simplifies to:
\[ 5x = \ln(2) \]
3. **Solve for \( x \).**
\[ x = \frac{\ln(2)}{5} \]
Thus, the exact solution for \( x \) is:
\[ x = \frac{\ln(2)}{5} \]
(Type your answer in the provided blue box as shown in the equation.)
**Diagram Explanation:**
There is no graph or diagram associated with this problem. The equation involves exponential and logarithmic properties which are transformed for solving \( x \).
**Note:**
Be sure to type the exact answer \( \frac{\ln(2)}{5} \) in decimal or fraction form as required by your assignment format.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
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, calculus and related others by exploring similar questions and additional content below.Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning