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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)
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.
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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