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 Logarithmic Equations
**Problem Statement:**
Solve for \( x \).
\[ \log_x 8 = 3 \]
In this problem, you are given the logarithmic equation \(\log_x 8 = 3\). Your task is to determine the value of \( x \).
**Explanation:**
The logarithmic equation \(\log_x 8 = 3\) can be rewritten in its exponential form to make it easier to solve. The equation \(\log_b a = c\) is equivalent to \(b^c = a\).
So, applying this rule to \(\log_x 8 = 3\):
\[
x^3 = 8
\]
To solve for \( x \), you need to take the cube root of both sides of the equation:
\[
x = \sqrt[3]{8}
\]
Since the cube root of 8 is 2, we get:
\[
x = 2
\]
**Answer:**
\[ x = 2 \]
You can verify your solution by substituting \( x \) back into the original logarithmic equation to see if both sides equal.
\[ \log_2 8 = 3 \]
Since \( 2^3 = 8 \), the solution is correct.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f516166-4390-4e41-acf6-d563f0703180%2Fd91a7e85-3e59-404c-be44-72f3215791d7%2Fvakuf3_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving Logarithmic Equations
**Problem Statement:**
Solve for \( x \).
\[ \log_x 8 = 3 \]
In this problem, you are given the logarithmic equation \(\log_x 8 = 3\). Your task is to determine the value of \( x \).
**Explanation:**
The logarithmic equation \(\log_x 8 = 3\) can be rewritten in its exponential form to make it easier to solve. The equation \(\log_b a = c\) is equivalent to \(b^c = a\).
So, applying this rule to \(\log_x 8 = 3\):
\[
x^3 = 8
\]
To solve for \( x \), you need to take the cube root of both sides of the equation:
\[
x = \sqrt[3]{8}
\]
Since the cube root of 8 is 2, we get:
\[
x = 2
\]
**Answer:**
\[ x = 2 \]
You can verify your solution by substituting \( x \) back into the original logarithmic equation to see if both sides equal.
\[ \log_2 8 = 3 \]
Since \( 2^3 = 8 \), the solution is correct.
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