Solve the Multi-Base Logarithm Equation: log₆₄ × log₉x = (log₆x² - log₆x)(log₉2.5 + log₉1.6)

To solve this problem, we'll carefully apply logarithmic properties:

• Step 1: Simplify the left-hand side:
  The left-hand side is given as $\log_64 \times \log_9x$. We simplify $\log_64$:
  $\log_64 = \frac{\log 4}{\log 6} = \frac{\log(2^2)}{\log 6} = \frac{2\log 2}{\log 6}$.
  Therefore, the left-hand side becomes $\frac{2\log 2}{\log 6} \times \log_9x$.
• Step 2: Simplify the right-hand side:
  The right-hand side is $(\log_6x^2 - \log_6x)(\log_92.5 + \log_91.6)$.
  First, simplify $\log_6x^2 - \log_6x = 2\log_6x - \log_6x = \log_6x$.
  For the other part, apply the product property: $\log_92.5 + \log_91.6 = \log_9(2.5 \times 1.6)$.
  Calculate $2.5 \times 1.6 = 4.0$, hence $\log_94$.
• Step 3: Equate and simplify:
  Now equate the simplified expressions: $\frac{2\log 2}{\log 6} \times \log_9x = \log_6x \cdot \log_94$.
  Change all logs to a common base (let's use natural log $\ln$) and solve:
• Step 4: Apply base conversion:
  $\log_9x = \frac{\ln x}{\ln 9}$, $\log_6x = \frac{\ln x}{\ln 6}$, and $\log_94 = \frac{\ln 4}{\ln 9}$.
• Step 5: Combine and solve:
  Perform algebraic manipulation and simplification:
  The equation becomes $\frac{2\ln 2}{\ln 6 \ln 9} \cdot \ln x = \frac{\ln x \cdot \ln 4}{\ln 6 \ln 9}$.
  Cancel $\ln x$ (non-zero due to $x > 0$) and solve for positive $x$.
• Conclude with the solution constraints:
  Given the properties and the domain involved, solution holds for all $0 < x$.

Therefore, the correct solution is: For all $0 < x$.