https://file.aiquickdraw.com/mathbot/file/1725494829030f9bx896t.jpg

The equation shown in the image is:

\[
\left( \frac{1}{9} \right)^{\frac{3x}{4}} = 2
\]

### Steps to solve:

1. **Rewrite the base as a power of 3**:
   \[
   \frac{1}{9} = 3^{-2}
   \]
   So the equation becomes:
   \[
   \left( 3^{-2} \right)^{\frac{3x}{4}} = 2
   \]

2. **Simplify the left-hand side**:
   When raising a power to another power, you multiply the exponents:
   \[
   3^{-2 \cdot \frac{3x}{4}} = 2
   \]
   This simplifies to:
   \[
   3^{-\frac{3x}{2}} = 2
   \]

3. **Take the natural logarithm of both sides**:
   \[
   \ln\left( 3^{-\frac{3x}{2}} \right) = \ln(2)
   \]
   Using the logarithmic identity \( \ln(a^b) = b\ln(a) \), the equation becomes:
   \[
   -\frac{3x}{2} \ln(3) = \ln(2)
   \]

4. **Solve for \( x \)**:
   \[
   x = \frac{2 \ln(2)}{-3 \ln(3)}
   \]

   Using approximate values for the natural logarithms:
   \[
   \ln(2) \approx 0.693, \quad \ln(3) \approx 1.099
   \]
   Plugging these into the equation:
   \[
   x = \frac{2 \cdot 0.693}{-3 \cdot 1.099} \approx \frac{1.386}{-3.297} \approx -0.42
   \]

Thus, the solution is approximately \( x \approx -0.42 \).

Would you like more details on any part of this solution?

### Related questions:
1. How do logarithmic properties apply to solving exponential equations?
2. Can this problem be solved without using logarithms?
3. What are the common logarithmic identities used in solving such problems?
4. How do approximate logarithmic values affect the final answer?
5. How would you handle this if the base were not expressible as a simple power?

### Tip:
When solving equations with exponents, always check if the base can be rewritten to simplify the equation.

Learn how to solve the exponential equation (1/9)^(3x/4) = 2 by rewriting the base, applying logarithmic properties, and solving for x. This problem demonstrates key concepts in exponential equations and logarithmic rules, suitable for high school students (Grades 9-12).

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