Find the value of the following `log_(cot30^@)tan45^@`

Answer

To find the value of \( \log_{\cot 30^\circ} \tan 45^\circ \), we can follow these steps:

### Step 1: Identify the values of \( \tan 45^\circ \) and \( \cot 30^\circ \)
- We know that:
\[ \tan 45^\circ = 1 \]
and
\[ \cot 30^\circ = \frac{1}{\tan 30^\circ} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3} \]

### Step 2: Rewrite the logarithm using the change of base formula
Using the change of base formula for logarithms, we can express \( \log_{\cot 30^\circ} \tan 45^\circ \) as:
\[ \log_{\cot 30^\circ} \tan 45^\circ = \frac{\log \tan 45^\circ}{\log \cot 30^\circ} \]

### Step 3: Substitute the known values into the logarithm
Now substituting the known values:
\[ \log_{\cot 30^\circ} \tan 45^\circ = \frac{\log 1}{\log \sqrt{3}} \]

### Step 4: Evaluate \( \log 1 \)
We know that:
\[ \log 1 = 0 \]

### Step 5: Substitute back into the expression
Now substituting back, we have:
\[ \log_{\cot 30^\circ} \tan 45^\circ = \frac{0}{\log \sqrt{3}} \]

### Step 6: Simplify the expression
Since the numerator is 0, the entire expression simplifies to:
\[ \log_{\cot 30^\circ} \tan 45^\circ = 0 \]

### Final Answer
Thus, the value of \( \log_{\cot 30^\circ} \tan 45^\circ \) is:
\[ \boxed{0} \]