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...
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![**Problem 3: Solve the Exponential Equation**
Solve the equation \(6^{2x-1} = 9\).
---
**Explanation for an Educational Website:**
To solve the exponential equation \(6^{2x-1} = 9\), follow these steps:
1. **Isolate the Exponential Part**: The equation is already in a form where the exponential expression is isolated.
2. **Apply Logarithms**: Use the logarithm to solve for \(x\). Taking the logarithm of both sides will allow you to bring the exponent down.
\[
\log(6^{2x-1}) = \log(9)
\]
3. **Use the Power Rule for Logarithms**: Apply the power rule, which states \( \log(a^b) = b \cdot \log(a) \).
\[
(2x-1) \cdot \log(6) = \log(9)
\]
4. **Solve for \(x\)**: Divide both sides by \(\log(6)\) to solve for the expression containing \(x\).
\[
2x-1 = \frac{\log(9)}{\log(6)}
\]
5. **Isolate \(x\)**: Add 1 to both sides and then divide by 2 to solve for \(x\).
\[
2x = \frac{\log(9)}{\log(6)} + 1
\]
\[
x = \frac{1}{2} \left(\frac{\log(9)}{\log(6)} + 1\right)
\]
6. **Calculate the Value**: Use a calculator to determine the numerical value of \(x\).
This process shows how to solve an exponential equation by using logarithms, demonstrating the application of mathematical properties to manipulate and solve expressions involving exponents.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F322a2375-fbba-45b8-8629-b5912bbd9c12%2Fa10daf56-4968-4305-9737-7d1aa4d237c8%2Fscbopvk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 3: Solve the Exponential Equation**
Solve the equation \(6^{2x-1} = 9\).
---
**Explanation for an Educational Website:**
To solve the exponential equation \(6^{2x-1} = 9\), follow these steps:
1. **Isolate the Exponential Part**: The equation is already in a form where the exponential expression is isolated.
2. **Apply Logarithms**: Use the logarithm to solve for \(x\). Taking the logarithm of both sides will allow you to bring the exponent down.
\[
\log(6^{2x-1}) = \log(9)
\]
3. **Use the Power Rule for Logarithms**: Apply the power rule, which states \( \log(a^b) = b \cdot \log(a) \).
\[
(2x-1) \cdot \log(6) = \log(9)
\]
4. **Solve for \(x\)**: Divide both sides by \(\log(6)\) to solve for the expression containing \(x\).
\[
2x-1 = \frac{\log(9)}{\log(6)}
\]
5. **Isolate \(x\)**: Add 1 to both sides and then divide by 2 to solve for \(x\).
\[
2x = \frac{\log(9)}{\log(6)} + 1
\]
\[
x = \frac{1}{2} \left(\frac{\log(9)}{\log(6)} + 1\right)
\]
6. **Calculate the Value**: Use a calculator to determine the numerical value of \(x\).
This process shows how to solve an exponential equation by using logarithms, demonstrating the application of mathematical properties to manipulate and solve expressions involving exponents.
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