Solve`(log_(3)x)(log_(5)9)- log_x 25 + log_(3) 2 = log_(3) 54`.

Explanation

To solve the equation \((\log_{3} x)(\log_{5} 9) - \log_{x} 25 + \log_{3} 2 = \log_{3} 54\), we will follow these steps:

### Step 1: Rewrite the logarithms using the change of base formula
Using the change of base formula, we can rewrite the logarithms:
\[ \log_{3} x = \frac{\log x}{\log 3}, \quad \log_{5} 9 = \frac{\log 9}{\log 5}, \quad \log_{x} 25 = \frac{\log 25}{\log x}, \quad \log_{3} 2 = \frac{\log 2}{\log 3}, \quad \log_{3} 54 = \frac{\log 54}{\log 3} \]
Substituting these into the equation gives:
\[ \left(\frac{\log x}{\log 3}\right) \left(\frac{\log 9}{\log 5}\right) - \frac{\log 25}{\log x} + \frac{\log 2}{\log 3} = \frac{\log 54}{\log 3} \]

### Step 2: Simplify the equation
Rearranging the equation, we have:
\[ \frac{\log x \cdot \log 9}{\log 3 \cdot \log 5} - \frac{\log 25}{\log x} + \frac{\log 2}{\log 3} = \frac{\log 54}{\log 3} \]
Next, we can multiply through by \(\log 3 \cdot \log x\) to eliminate the denominators:
\[ \log x \cdot \log 9 - \log 25 \cdot \log 3 + \log 2 \cdot \log x = \log 54 \cdot \log x \]

### Step 3: Combine like terms
Rearranging gives:
\[ \log x \cdot \log 9 + \log 2 \cdot \log x - \log 54 \cdot \log x = \log 25 \cdot \log 3 \]
Factoring out \(\log x\):
\[ \log x (\log 9 + \log 2 - \log 54) = \log 25 \cdot \log 3 \]

### Step 4: Simplify the logarithmic expression
Using the property \(\log a + \log b = \log(ab)\) and \(\log a - \log b = \log\left(\frac{a}{b}\right)\):
\[ \log x \left(\log(18) - \log(54)\right) = \log 25 \cdot \log 3 \]
Thus,
\[ \log x \cdot \log\left(\frac{18}{54}\right) = \log 25 \cdot \log 3 \]
This simplifies to:
\[ \log x \cdot \log\left(\frac{1}{3}\right) = \log 25 \cdot \log 3 \]

### Step 5: Solve for \(\log x\)
Dividing both sides by \(\log\left(\frac{1}{3}\right)\):
\[ \log x = \frac{\log 25 \cdot \log 3}{\log\left(\frac{1}{3}\right)} \]
Since \(\log\left(\frac{1}{3}\right) = -\log 3\):
\[ \log x = -\frac{\log 25}{1} = -\log 25 \]
Thus,
\[ x = 10^{-\log 25} = \frac{1}{25} \]

### Step 6: Final answer
The solution to the equation is:
\[ x = \frac{1}{25} \]