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The equation you provided is \( 8^x = \frac{1}{128} \). Let's solve it analytically.

### Step 1: Express both sides with the same base.
We know that:
- \( 8 = 2^3 \), so \( 8^x = (2^3)^x = 2^{3x} \).
- Also, \( 128 = 2^7 \), so \( \frac{1}{128} = 2^{-7} \).

Thus, the equation becomes:
\[
2^{3x} = 2^{-7}
\]

### Step 2: Set the exponents equal.
Since the bases are the same, we can equate the exponents:
\[
3x = -7
\]

### Step 3: Solve for \( x \).
\[
x = \frac{-7}{3}
\]

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

Would you like any further details or have any questions on this solution?

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Here are 5 related questions to expand on the topic:
1. How would the solution change if the equation were \( 8^x = 128 \)?
2. What is the general method for solving exponential equations with different bases?
3. How would you solve an equation like \( 9^x = \frac{1}{81} \)?
4. Can you express \( 8^x = 1 \) in logarithmic form and solve for \( x \)?
5. How do you solve equations involving fractional exponents like \( 8^{x/2} = \frac{1}{128} \)?

**Tip**: When solving exponential equations, it's often helpful to express both sides using the same base for easier manipulation of exponents.

Solve the equation analytically: 8^x = 1/128

Learn how to solve the exponential equation 8^x = 1/128 using base conversion and equating exponents. This detailed step-by-step solution helps understand the application of exponent laws and theorems. Ideal for students in Grades 9-12.

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