Question
Solve the exponential equation. Express irrational solutions in exact form. \[ 8^{x-7}=64 \]
Ask by Turnbull Pena. in the United States
Jan 22,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solution is \( x = 9 \).
Solution
To solve the exponential equation:
\[
8^{x-7} = 64
\]
**Step 1: Express both sides with the same base.**
We know that:
- \(8 = 2^3\)
- \(64 = 2^6\)
Rewrite the equation using base 2:
\[
(2^3)^{x-7} = 2^6
\]
**Step 2: Simplify the left side using the power of a power rule \((a^m)^n = a^{m \cdot n}\):**
\[
2^{3(x-7)} = 2^6
\]
**Step 3: Since the bases are the same, set the exponents equal to each other:**
\[
3(x - 7) = 6
\]
**Step 4: Solve for \(x\):**
\[
3x - 21 = 6
\]
\[
3x = 27
\]
\[
x = 9
\]
**Conclusion:**
The solution to the equation is:
\[
\boxed{9}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
To solve the equation \( 8^{x-7} = 64 \), we can express both sides with the same base. We can rewrite \( 8 \) and \( 64 \) as powers of \( 2 \): \[ 8 = 2^3 \quad \text{and} \quad 64 = 2^6. \] Now, substituting these values into the equation gives us: \[ (2^3)^{x-7} = 2^6. \] Applying the power of a power property, we have: \[ 2^{3(x-7)} = 2^6. \] Since the bases are the same, we can equate the exponents: \[ 3(x-7) = 6. \] Now, solving for \( x \): \[ 3x - 21 = 6 \\ 3x = 6 + 21 \\ 3x = 27 \\ x = \frac{27}{3} \\ x = 9. \] Thus, the solution to the equation is: \[ \boxed{9}. \]
