Use logarithms to solve the equation for a. 100¹ = 1,000 x= [blank] Enter your answer as an integer or a fraction. If your answer is a fraction, enter it like this: 3/14
Use logarithms to solve the equation for a. 100¹ = 1,000 x= [blank] Enter your answer as an integer or a fraction. If your answer is a fraction, enter it like this: 3/14
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,...
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Question
![**Solving Exponential Equations Using Logarithms**
**Equation to Solve:**
\[ 100^{4x} = 1,000 \]
**Objective:**
Find the value of \( x \).
**Method:**
Utilize logarithms to solve the exponential equation for \( x \).
**Solution Steps:**
1. Take the logarithm of both sides of the equation:
\[ \log(100^{4x}) = \log(1,000) \]
2. Apply the power rule of logarithms:
\[ 4x \cdot \log(100) = \log(1,000) \]
3. Calculate the logarithm values:
- \( \log(100) = 2 \) because \( 100 = 10^2 \)
- \( \log(1,000) = 3 \) because \( 1,000 = 10^3 \)
4. Substitute the values back into the equation:
\[ 4x \cdot 2 = 3 \]
5. Solve for \( x \):
\[ 8x = 3 \]
\[ x = \frac{3}{8} \]
**Answer Format:**
Enter your answer as an integer or a fraction. If your answer is a fraction, input it in the format: 3/14.
---
**Final Answer:**
\[ x = \frac{3}{8} \]
By solving this equation, we learn how to apply logarithmic rules to solve for unknown variables in exponential equations, enhancing our mathematical problem-solving skills.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F86f7f240-52bf-4bdf-b932-d5abd63fff02%2Faca8253f-8dac-4a7d-942c-77bf1f3db34a%2Fsmjonw7_processed.png&w=3840&q=75)
Transcribed Image Text:**Solving Exponential Equations Using Logarithms**
**Equation to Solve:**
\[ 100^{4x} = 1,000 \]
**Objective:**
Find the value of \( x \).
**Method:**
Utilize logarithms to solve the exponential equation for \( x \).
**Solution Steps:**
1. Take the logarithm of both sides of the equation:
\[ \log(100^{4x}) = \log(1,000) \]
2. Apply the power rule of logarithms:
\[ 4x \cdot \log(100) = \log(1,000) \]
3. Calculate the logarithm values:
- \( \log(100) = 2 \) because \( 100 = 10^2 \)
- \( \log(1,000) = 3 \) because \( 1,000 = 10^3 \)
4. Substitute the values back into the equation:
\[ 4x \cdot 2 = 3 \]
5. Solve for \( x \):
\[ 8x = 3 \]
\[ x = \frac{3}{8} \]
**Answer Format:**
Enter your answer as an integer or a fraction. If your answer is a fraction, input it in the format: 3/14.
---
**Final Answer:**
\[ x = \frac{3}{8} \]
By solving this equation, we learn how to apply logarithmic rules to solve for unknown variables in exponential equations, enhancing our mathematical problem-solving skills.
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