The equation is:

\[
8^{x-1} = 16
\]

### Step 1: Express both sides as powers of 2.
We know:
\[
8 = 2^3 \quad \text{and} \quad 16 = 2^4
\]

So, rewrite the equation using powers of 2:
\[
(2^3)^{x-1} = 2^4
\]

### Step 2: Simplify the exponents.
Apply the power of a power rule \((a^m)^n = a^{mn}\):
\[
2^{3(x-1)} = 2^4
\]

This simplifies to:
\[
2^{3x - 3} = 2^4
\]

### Step 3: Set the exponents equal.
Since the bases are the same, we can set the exponents equal:
\[
3x - 3 = 4
\]

### Step 4: Solve for \(x\).
Now solve the linear equation:
\[
3x - 3 = 4
\]

Add 3 to both sides:
\[
3x = 7
\]

Divide by 3:
\[
x = \frac{7}{3}
\]

### Final Answer:
\[
x = \frac{7}{3}
\]

Would you like more details on any step or have any questions?

Here are 5 related questions:
1. How do you solve exponential equations when the bases are different?
2. What are the laws of exponents and how do they apply to this problem?
3. How can you solve equations involving logarithms?
4. What are some common techniques for simplifying powers?
5. Can you express any number as a power of a different base?

**Tip:** When solving exponential equations, always try to express both sides with the same base to make solving easier!

Learn how to solve the exponential equation 8^x - 1 = 16 with a detailed step-by-step solution. This problem explores key concepts like rewriting bases as powers of 2 and using the power of a power rule. Suitable for students in Grades 9-11.

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