The value of `(log_(3) 5 xx log_(25) 27 xx log_(49) 7)/(log_(81)3)` is

Options

A. 1
B. 6
C. `(2)/(3)`
D. 3

Answer

To solve the expression \((\log_{3} 5 \times \log_{25} 27 \times \log_{49} 7) / (\log_{81} 3)\), we can use the change of base formula and properties of logarithms. Let's break it down step by step.

### Step 1: Change of Base Formula
Using the change of base formula, we can express each logarithm in terms of natural logarithms (or any common base). The formula is:
\[ \log_{a} b = \frac{\log b}{\log a} \]

Applying this to our expression:
\[ \log_{3} 5 = \frac{\log 5}{\log 3}, \quad \log_{25} 27 = \frac{\log 27}{\log 25}, \quad \log_{49} 7 = \frac{\log 7}{\log 49}, \quad \log_{81} 3 = \frac{\log 3}{\log 81} \]

### Step 2: Substitute into the Expression
Substituting these into the original expression gives:
\[ \frac{\left(\frac{\log 5}{\log 3}\right) \times \left(\frac{\log 27}{\log 25}\right) \times \left(\frac{\log 7}{\log 49}\right)}{\frac{\log 3}{\log 81}} \]

### Step 3: Simplify the Expression
This simplifies to:
\[ \frac{\log 5 \times \log 27 \times \log 7}{\log 3 \times \log 25} \times \frac{\log 81}{\log 3} \]

### Step 4: Further Simplification
Now, we can simplify \(\log 25\) and \(\log 81\):
\[ \log 25 = \log(5^2) = 2 \log 5 \]
\[ \log 81 = \log(3^4) = 4 \log 3 \]

Substituting these back into our expression gives:
\[ \frac{\log 5 \times \log 27 \times \log 7}{\log 3 \times 2 \log 5} \times \frac{4 \log 3}{\log 3} \]

### Step 5: Canceling Terms
Notice that \(\log 5\) in the numerator and denominator cancels out:
\[ \frac{\log 27 \times \log 7}{2} \times 4 \]
This simplifies to:
\[ 2 \log 27 \times \log 7 \]

### Step 6: Further Simplification of \(\log 27\)
Since \(\log 27 = \log(3^3) = 3 \log 3\), we have:
\[ 2 \times 3 \log 3 \times \log 7 = 6 \log 3 \times \log 7 \]

### Step 7: Final Calculation
Now we need to evaluate \(6 \log 3 \times \log 7\). However, we can also express \(\log 7\) in terms of base 3 if needed or keep it as is.

### Conclusion
The final value of the expression is:
\[ \frac{6 \log 3 \times \log 7}{\log 3} = 6 \log 7 \]

Thus, the value of the original expression is \(3\).