The value of ` (6 a^(log_(e)b)(log_(a^(2))b)(log_(b^(2))a))/(e^(log_(e)a*log_(e)b))` is

Options

A. independent of a
B. independent of b
C. dependent on a
D. dependent on b

Explanation

To solve the expression \( \frac{6 a^{\log_{e} b} \cdot \log_{a^{2}} b \cdot \log_{b^{2}} a}{e^{\log_{e} a \cdot \log_{e} b}} \), we will simplify it step by step using properties of logarithms.

### Step 1: Rewrite the logarithms
Using the change of base formula, we can express the logarithms in terms of natural logarithms:
\[ \log_{a^2} b = \frac{\log_{e} b}{\log_{e} a^2} = \frac{\log_{e} b}{2 \log_{e} a} \]
\[ \log_{b^2} a = \frac{\log_{e} a}{\log_{e} b^2} = \frac{\log_{e} a}{2 \log_{e} b} \]

### Step 2: Substitute the rewritten logarithms into the expression
Now substituting these back into the original expression:
\[ \frac{6 a^{\log_{e} b} \cdot \frac{\log_{e} b}{2 \log_{e} a} \cdot \frac{\log_{e} a}{2 \log_{e} b}}{e^{\log_{e} a \cdot \log_{e} b}} \]

### Step 3: Simplify the expression
The logarithms in the numerator simplify:
\[ \frac{6 a^{\log_{e} b} \cdot \frac{\log_{e} b}{2 \log_{e} a} \cdot \frac{\log_{e} a}{2 \log_{e} b}}{e^{\log_{e} a \cdot \log_{e} b}} = \frac{6 a^{\log_{e} b} \cdot \frac{1}{4}}{e^{\log_{e} a \cdot \log_{e} b}} = \frac{3}{2} \cdot \frac{a^{\log_{e} b}}{e^{\log_{e} a \cdot \log_{e} b}} \]

### Step 4: Simplify the denominator
Using the property \( e^{\log_{e} x} = x \):
\[ e^{\log_{e} a \cdot \log_{e} b} = a^{\log_{e} b} \]

### Step 5: Final simplification
Now we can simplify the expression:
\[ \frac{3}{2} \cdot \frac{a^{\log_{e} b}}{a^{\log_{e} b}} = \frac{3}{2} \]

### Conclusion
Thus, the value of the given expression is:
\[ \frac{3}{2} \]