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
Question
![**Using Logarithmic Properties to Solve Equations**
In this section, we will explore how to solve the equation using the properties of logarithmic functions.
Given:
\[ \log_5(x^2 + 1) - \log_5(x - 1) = 1 \]
Refer to the definition (p. 75) and properties of logarithmic functions as explained in Theorem 1 (p. 77) to solve this problem.
### Steps to Solve:
1. **Apply the Quotient Rule**:
Using the quotient rule for logarithms: \(\log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right)\).
\[
\log_5 \left(\frac{x^2 + 1}{x - 1}\right) = 1
\]
2. **Convert to Exponential Form**:
If \(\log_b(a) = c\), then \(a = b^c\).
\[
\frac{x^2 + 1}{x - 1} = 5^1
\]
3. **Solve the Equation**:
\[
\frac{x^2 + 1}{x - 1} = 5
\]
Multiply both sides by \(x - 1\):
\[
x^2 + 1 = 5(x - 1)
\]
Simplify and solve for \(x\):
\[
x^2 + 1 = 5x - 5
\]
Rearrange:
\[
x^2 - 5x + 6 = 0
\]
4. **Factor the Quadratic**:
\[(x - 2)(x - 3) = 0\]
Solve for \(x\):
\[
x = 2 \quad \text{or} \quad x = 3
\]
5. **Check for Validity**:
Both solutions need to satisfy the original equation conditions:
- For \(x = 2\), calculate the log terms:
\[
\log_5((2)^2 + 1) - \log_5(2 - 1) = 1
\]
\[
\log_5(5)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F236de4c3-f237-402b-b0e0-efa1a8147b53%2Fd95c1a7c-a404-4d4c-b80b-08513c1673c5%2Fnnx6ish_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Using Logarithmic Properties to Solve Equations**
In this section, we will explore how to solve the equation using the properties of logarithmic functions.
Given:
\[ \log_5(x^2 + 1) - \log_5(x - 1) = 1 \]
Refer to the definition (p. 75) and properties of logarithmic functions as explained in Theorem 1 (p. 77) to solve this problem.
### Steps to Solve:
1. **Apply the Quotient Rule**:
Using the quotient rule for logarithms: \(\log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right)\).
\[
\log_5 \left(\frac{x^2 + 1}{x - 1}\right) = 1
\]
2. **Convert to Exponential Form**:
If \(\log_b(a) = c\), then \(a = b^c\).
\[
\frac{x^2 + 1}{x - 1} = 5^1
\]
3. **Solve the Equation**:
\[
\frac{x^2 + 1}{x - 1} = 5
\]
Multiply both sides by \(x - 1\):
\[
x^2 + 1 = 5(x - 1)
\]
Simplify and solve for \(x\):
\[
x^2 + 1 = 5x - 5
\]
Rearrange:
\[
x^2 - 5x + 6 = 0
\]
4. **Factor the Quadratic**:
\[(x - 2)(x - 3) = 0\]
Solve for \(x\):
\[
x = 2 \quad \text{or} \quad x = 3
\]
5. **Check for Validity**:
Both solutions need to satisfy the original equation conditions:
- For \(x = 2\), calculate the log terms:
\[
\log_5((2)^2 + 1) - \log_5(2 - 1) = 1
\]
\[
\log_5(5)
Expert Solution
Step 1
We need to find value of x.
Step by step
Solved in 2 steps with 2 images
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