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The solutions to the equation \( \log_3(9x) + \log_3\left(\frac{x^5}{81}\right) = 3 \) include real and complex values.

The real solution is:

\[
x = 3^{5/6}
\]

This can be approximated numerically as:

\[
x \approx 2.049
\]

The other solutions are complex, but typically, for logarithmic equations like this, we focus on the real solution.

Would you like a more detailed explanation of the steps involved, or have any questions?

### Related Questions:
1. How do we handle logarithmic properties when solving equations?
2. Can this logarithmic equation be simplified differently?
3. How do complex solutions arise in logarithmic equations?
4. What is the numerical value of \( 3^{5/6} \) to a higher precision?
5. Are there restrictions on the domain of \( x \) for logarithmic equations?

**Tip:** When solving logarithmic equations, always check for extraneous solutions by considering the domain restrictions of the logarithmic functions.

Given that \(\log_3(9x) + \log_3\left(\frac{x^5}{81}\right) = 3\), solve for x.

Learn how to solve the logarithmic equation \( \log_3(9x) + \log_3\left(\frac{x^5}{81}\right) = 3 \). This problem explores the properties of logarithms and exponents and provides a step-by-step approach to solving for x.

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