The value of `(log_5 9*log_7 5*log_3 7)/(log_3 sqrt(6))+1/(log_4 sqrt(6))` is co-prime with

Options

A. 1
B. 3
C. 4
D. 5

Answer

To solve the given expression \((\log_5 9 \cdot \log_7 5 \cdot \log_3 7) / \log_3 \sqrt{6} + 1 / \log_4 \sqrt{6}\), we will simplify it step by step.

### Step 1: Rewrite the logarithms using properties
We can use the change of base formula and the power property of logarithms. Recall that \(\log_b a = \frac{\log_k a}{\log_k b}\) for any base \(k\), and \(\log_b (a^n) = n \log_b a\).

\[ \log_5 9 = \frac{\log 9}{\log 5}, \quad \log_7 5 = \frac{\log 5}{\log 7}, \quad \log_3 7 = \frac{\log 7}{\log 3} \]

Substituting these into the expression gives:

\[ \frac{\left(\frac{\log 9}{\log 5}\right) \cdot \left(\frac{\log 5}{\log 7}\right) \cdot \left(\frac{\log 7}{\log 3}\right)}{\log_3 \sqrt{6}} + \frac{1}{\log_4 \sqrt{6}} \]

### Step 2: Simplify the product in the numerator
Notice that the \(\log 5\) and \(\log 7\) terms cancel out:

\[ = \frac{\log 9}{\log 3} \cdot \frac{1}{\log_3 \sqrt{6}} + \frac{1}{\log_4 \sqrt{6}} \]

### Step 3: Simplify \(\log_3 \sqrt{6}\) and \(\log_4 \sqrt{6}\)
Using the power property of logarithms:

\[ \log_3 \sqrt{6} = \log_3 (6^{1/2}) = \frac{1}{2} \log_3 6 \]
\[ \log_4 \sqrt{6} = \log_4 (6^{1/2}) = \frac{1}{2} \log_4 6 \]

Substituting these back into the expression gives:

\[ = \frac{\log 9}{\log 3} \cdot \frac{2}{\log 6} + \frac{2}{\log 6} \]

### Step 4: Combine the terms
Factoring out \(\frac{2}{\log 6}\):

\[ = \frac{2}{\log 6} \left(\frac{\log 9}{\log 3} + 1\right) \]

### Step 5: Simplify \(\frac{\log 9}{\log 3}\)
We know that \(\log 9 = \log(3^2) = 2 \log 3\):

\[ \frac{\log 9}{\log 3} = 2 \]

Thus, substituting this back gives:

\[ = \frac{2}{\log 6} (2 + 1) = \frac{2 \cdot 3}{\log 6} = \frac{6}{\log 6} \]

### Step 6: Evaluate \(\frac{6}{\log 6}\)
Now we need to find if \(6\) is co-prime with any of the given options. The only common factor of \(6\) is \(1\), so we check the options.

### Conclusion
The value \(6\) is co-prime with \(1\), \(3\), and \(5\). Therefore, the answer is:

**Coprime with:** \(1, 3, 5\)