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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Question
![**Problem Statement**:
Use like bases to solve:
\( 27 \cdot 3^{2x} = 243 \)
**Solution**:
To solve this equation, express all numbers with the same base:
- \(27\) can be written as \(3^3\).
- \(243\) can be written as \(3^5\).
Substitute these into the equation:
\[ 3^3 \cdot 3^{2x} = 3^5 \]
Combine the exponents on the left-hand side:
\[ 3^{3 + 2x} = 3^5 \]
Since the bases are the same, set the exponents equal:
\[ 3 + 2x = 5 \]
Solve for \(x\):
\[ 2x = 5 - 3 \]
\[ 2x = 2 \]
\[ x = 1 \]
**Answer**:
The solution is \( x = 1 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbf8e267e-10e5-42d7-8fd7-a8f3672a74ed%2F21d1eca8-7744-4827-82ba-db4dab23812f%2Fb5o3p9j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement**:
Use like bases to solve:
\( 27 \cdot 3^{2x} = 243 \)
**Solution**:
To solve this equation, express all numbers with the same base:
- \(27\) can be written as \(3^3\).
- \(243\) can be written as \(3^5\).
Substitute these into the equation:
\[ 3^3 \cdot 3^{2x} = 3^5 \]
Combine the exponents on the left-hand side:
\[ 3^{3 + 2x} = 3^5 \]
Since the bases are the same, set the exponents equal:
\[ 3 + 2x = 5 \]
Solve for \(x\):
\[ 2x = 5 - 3 \]
\[ 2x = 2 \]
\[ x = 1 \]
**Answer**:
The solution is \( x = 1 \).
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