To solve the equation \( 3^x = \frac{1}{9} \), follow these steps:

1. **Express 9 as a power of 3**: 
   \[
   9 = 3^2
   \]
   So the equation becomes:
   \[
   3^x = \frac{1}{3^2}
   \]

2. **Rewrite the fraction as a negative exponent**:
   \[
   \frac{1}{3^2} = 3^{-2}
   \]
   Now, the equation is:
   \[
   3^x = 3^{-2}
   \]

3. **Since the bases are the same, set the exponents equal to each other**:
   \[
   x = -2
   \]

So, the solution is \( x = -2 \).

Would you like more details, or do you have any questions? Here are 5 related questions you might find interesting:

1. How would you solve \( 2^x = \frac{1}{8} \)?
2. Can you solve \( 3^{2x} = 81 \)?
3. What happens if the equation was \( 3^x = 27 \)?
4. How would you approach solving \( 4^x = 16 \) without a calculator?
5. What are the properties of exponents that are useful in solving such equations?

**Tip:** Remember that if you have an equation like \( a^x = a^y \), you can directly set the exponents equal to each other: \( x = y \).

Learn how to solve the exponential equation 3^x = 1/9 with a detailed step-by-step solution. This problem involves understanding exponents and negative exponents. Suitable for students in Grades 9-12.

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