Step 1: Understanding the Question:
This is a logarithmic equation that requires simplifying properties of logs, specifically the change of base and power rules.
Step 2: Key Formula or Approach:
Use \( \log_a b^n = n \log_a b \), \( \log_{a^m} b = \frac{1}{m} \log_a b \), and let \( \log_3 x = t \).
Step 3: Detailed Explanation:
Term 1: \( \sqrt{\log_3 x^{16}} = \sqrt{16 \log_3 x} = 4\sqrt{\log_3 x} \).
Term 2: \( 9\log_{27} \sqrt[3]{\frac{3}{x}} = 9\log_{3^3} \left(\frac{3}{x}\right)^{1/3} = 9 \cdot \frac{1}{3} \cdot \frac{1}{3} \log_3 \left(\frac{3}{x}\right) \).
\( = 1 \cdot (\log_3 3 - \log_3 x) = 1 - \log_3 x \).
Let \( \sqrt{\log_3 x} = y \), so \( \log_3 x = y^2 \).
Substituting into the equation:
\( 4y + 1 - y^2 = 5 \)
\( y^2 - 4y + 4 = 0 \)
\( (y - 2)^2 = 0 \Rightarrow y = 2 \).
Substituting back:
\( \sqrt{\log_3 x} = 2 \Rightarrow \log_3 x = 4 \).
\( x = 3^4 = 81 \).
Step 4: Final Answer:
The value of \( x \) is 81.