`log{log_(ab)a+(1)/(log_(b)ab)}` =

Options

A. `0`
B. `1`
C. log ab
D. none of these

Answer

To solve the expression \( \log\left(\log_{ab} a + \frac{1}{\log_{b} ab}\right) \), we will follow these steps:

### Step 1: Rewrite the logarithmic terms
We start with the expression:
\[ \log_{ab} a + \frac{1}{\log_{b} ab} \]
Using the change of base formula, we can rewrite \( \log_{ab} a \) as:
\[ \log_{ab} a = \frac{\log a}{\log(ab)} = \frac{\log a}{\log a + \log b} \]

### Step 2: Rewrite the second term
Next, we rewrite \( \log_{b} ab \):
\[ \log_{b} ab = \frac{\log(ab)}{\log b} = \frac{\log a + \log b}{\log b} \]
Thus, we can find \( \frac{1}{\log_{b} ab} \):
\[ \frac{1}{\log_{b} ab} = \frac{\log b}{\log a + \log b} \]

### Step 3: Combine the two terms
Now we can combine both terms:
\[ \log_{ab} a + \frac{1}{\log_{b} ab} = \frac{\log a}{\log a + \log b} + \frac{\log b}{\log a + \log b} \]
Since both fractions have the same denominator, we can add them:
\[ = \frac{\log a + \log b}{\log a + \log b} = 1 \]

### Step 4: Take the logarithm of the result
Now we take the logarithm of the result:
\[ \log(1) \]
Since the logarithm of 1 is 0 for any base, we have:
\[ \log(1) = 0 \]

### Final Answer
Thus, the final answer is:
\[ \log\left(\log_{ab} a + \frac{1}{\log_{b} ab}\right) = 0 \]