Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter4: Exponential And Logarithmic Functions
Section4.CT: Chapter Test
Problem 11CT
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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.
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What is the step by step solution to solve this? I am having hard time understanding the steps.
![**Problem 2: Solve the Equation for x**
\[ 2 \ln(x-3) = \ln(x+5) + \ln 4 \]
This is a logarithmic equation where you need to solve for the variable \( x \).
Steps to solve:
1. **Combine Logarithmic Terms:**
The equation can be rewritten by using logarithmic properties. The right side, \(\ln(x+5) + \ln 4\), can be combined into a single logarithm:
\[ \ln((x+5) \cdot 4) = \ln(4(x+5)) \]
2. **Set the Logarithms Equal:**
Now the equation becomes:
\[ 2 \ln(x-3) = \ln(4(x+5)) \]
3. **Eliminate the Logarithms:**
Since the bases are the same, you can set the arguments equal to each other:
\[ (x-3)^2 = 4(x+5) \]
4. **Expand and Simplify the Equation:**
Expand and rearrange the equation to solve for \( x \).
5. **Solve the Quadratic Equation:**
Solve for \( x \) considering the domain constraints due to the logarithms.
**Note:** Ensure that the values obtained for \( x \) satisfy the original equation’s requirements, specifically that the arguments of the logarithms are positive.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a799525-bc02-4aa7-a724-19d6c1834605%2Fd3a89502-156d-4c83-9c04-9b21baceea21%2F92uecc5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 2: Solve the Equation for x**
\[ 2 \ln(x-3) = \ln(x+5) + \ln 4 \]
This is a logarithmic equation where you need to solve for the variable \( x \).
Steps to solve:
1. **Combine Logarithmic Terms:**
The equation can be rewritten by using logarithmic properties. The right side, \(\ln(x+5) + \ln 4\), can be combined into a single logarithm:
\[ \ln((x+5) \cdot 4) = \ln(4(x+5)) \]
2. **Set the Logarithms Equal:**
Now the equation becomes:
\[ 2 \ln(x-3) = \ln(4(x+5)) \]
3. **Eliminate the Logarithms:**
Since the bases are the same, you can set the arguments equal to each other:
\[ (x-3)^2 = 4(x+5) \]
4. **Expand and Simplify the Equation:**
Expand and rearrange the equation to solve for \( x \).
5. **Solve the Quadratic Equation:**
Solve for \( x \) considering the domain constraints due to the logarithms.
**Note:** Ensure that the values obtained for \( x \) satisfy the original equation’s requirements, specifically that the arguments of the logarithms are positive.
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