Question
Solve the equation. \( 3^{x-5}=243 \) \( x=[?] \)
Ask by Craig Davey. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x = 10 \)
Solution
To solve the equation:
\[
3^{x-5} = 243
\]
**Step 1: Express 243 as a Power of 3**
First, recognize that \( 243 \) can be written as a power of \( 3 \):
\[
243 = 3^5
\]
**Step 2: Rewrite the Original Equation**
Substitute \( 243 \) with \( 3^5 \) in the original equation:
\[
3^{x-5} = 3^5
\]
**Step 3: Set the Exponents Equal**
Since the bases are the same and the equation holds true, the exponents must be equal:
\[
x - 5 = 5
\]
**Step 4: Solve for \( x \)**
Add \( 5 \) to both sides of the equation to isolate \( x \):
\[
x = 5 + 5
\]
\[
x = 10
\]
**Final Answer:**
\[
x = 10
\]
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Simplify this solution Extra Insights
To solve the equation \( 3^{x-5} = 243 \), we can start by expressing 243 as a power of 3. We know that \( 243 = 3^5 \), so we can rewrite the equation as: \( 3^{x-5} = 3^5 \) Since the bases are the same, we can set the exponents equal to each other: \( x - 5 = 5 \) Now, solving for \( x \): \( x = 5 + 5 = 10 \) So, the solution is \( x = 10 \).
