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

Options

A. `3 y^(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 equation
We start with the given equation:
\[ \log_3 x \cdot \log_y 3 \cdot \log_2 y = 5 \]

### Step 2: Isolate \( \log_3 x \)
To isolate \( \log_3 x \), we can express it in terms of the other logarithms:
\[ \log_3 x = \frac{5}{\log_y 3 \cdot \log_2 y} \]

### Step 3: Use the change of base formula
Using the change of base formula, we can express \( \log_y 3 \) and \( \log_2 y \):
\[ \log_y 3 = \frac{\log_3 3}{\log_3 y} = \frac{1}{\log_3 y} \]
\[ \log_2 y = \frac{\log_3 y}{\log_3 2} \]

### Step 4: Substitute back into the equation
Substituting these into our isolated equation for \( \log_3 x \):
\[ \log_3 x = \frac{5}{\left(\frac{1}{\log_3 y}\right) \cdot \left(\frac{\log_3 y}{\log_3 2}\right)} = \frac{5 \cdot \log_3 2}{1} = 5 \log_3 2 \]

### Step 5: Exponentiate to solve for \( x \)
Now we exponentiate both sides to solve for \( x \):
\[ x = 3^{5 \log_3 2} \]

### Step 6: Simplify using properties of exponents
Using the property \( a^{\log_a b} = b \):
\[ x = 2^5 \]

### Step 7: Calculate \( 2^5 \)
Calculating \( 2^5 \):
\[ x = 32 \]

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

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