If `log_(3)xlog_(y)3log_(2)y=5`, then x=

Options

A. `3y^(5)`
B. `243`
C. `32`
D. none of these

Answer

To solve the equation \( \log_3 x \cdot \log_y 3 \cdot \log_2 y = 5 \), we will follow these steps:

### Step 1: Rewrite the logarithms using the change of base formula
Using the change of base formula, we can express each logarithm in terms of natural logarithms (or any common base):
\[ \log_3 x = \frac{\log_e x}{\log_e 3}, \quad \log_y 3 = \frac{\log_e 3}{\log_e y}, \quad \log_2 y = \frac{\log_e y}{\log_e 2} \]

### Step 2: Substitute the expressions into the equation
Substituting these expressions into the original equation gives:
\[ \left(\frac{\log_e x}{\log_e 3}\right) \cdot \left(\frac{\log_e 3}{\log_e y}\right) \cdot \left(\frac{\log_e y}{\log_e 2}\right) = 5 \]

### Step 3: Simplify the equation
Notice that \( \log_e 3 \) in the numerator of the second term and in the denominator of the first term cancels out, as does \( \log_e y \):
\[ \frac{\log_e x}{\log_e 2} = 5 \]

### Step 4: Rewrite the equation in logarithmic form
Now we can rewrite the equation in logarithmic form:
\[ \log_2 x = 5 \]

### Step 5: Convert to exponential form
To find \( x \), we convert the logarithmic equation back to exponential form:
\[ x = 2^5 \]

### Step 6: Calculate the value of \( x \)
Calculating \( 2^5 \) gives:
\[ x = 32 \]

Thus, the value of \( x \) is \( 32 \).

### Final Answer
\[ \boxed{32} \]