Question
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Tutor-Verified Answer
Answer
\( x = -3 \)
Solution
Solve the equation \( 5^{x-1}=\frac{1}{625} \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(5^{x-1}=\frac{1}{625}\)
- step1: Rewrite in exponential form:
\(5^{x-1}=5^{-4}\)
- step2: Set the exponents equal:
\(x-1=-4\)
- step3: Move the constant to the right side:
\(x=-4+1\)
- step4: Add the numbers:
\(x=-3\)
The solution to the equation \(5^{x-1}=\frac{1}{625}\) is \(x=-3\).
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Simplify this solution The Deep Dive
To solve \( 5^{x-1}=\frac{1}{625} \), we can express \( 625 \) as a power of \( 5 \). Since \( 625 = 5^4 \), we have \( \frac{1}{625} = 5^{-4} \). Therefore, the equation transforms into: \[ 5^{x-1} = 5^{-4} \] Since the bases are the same, we can set the exponents equal to each other: \[ x - 1 = -4 \] Adding \( 1 \) to both sides gives: \[ x = -3 \] So the solution is \( x = -3 \).
