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

Options

A. `2`
B. `1`
C. `6`
D. `3`

Answer

To solve the expression \(\frac{\log_{3}5 \times \log_{25}27 \times \log_{49}7}{\log_{81}3}\), we will simplify each logarithm step by step.

### Step 1: Change of Base Formula
Using the change of base formula, we can rewrite each logarithm:
\[ \log_{a}b = \frac{\log_{c}b}{\log_{c}a} \]
Let's use base 3 for our calculations.

1. \(\log_{3}5 = \log_{3}5\)
2. \(\log_{25}27 = \frac{\log_{3}27}{\log_{3}25}\)
3. \(\log_{49}7 = \frac{\log_{3}7}{\log_{3}49}\)
4. \(\log_{81}3 = \frac{\log_{3}3}{\log_{3}81}\)

### Step 2: Substitute the Values
Substituting these into the original expression gives:
\[ \frac{\log_{3}5 \times \frac{\log_{3}27}{\log_{3}25} \times \frac{\log_{3}7}{\log_{3}49}}{\frac{\log_{3}3}{\log_{3}81}} \]

### Step 3: Simplify the Expression
This can be rearranged to:
\[ \frac{\log_{3}5 \times \log_{3}27 \times \log_{3}7}{\log_{3}25 \times \log_{3}49} \times \frac{\log_{3}81}{\log_{3}3} \]

### Step 4: Simplify Each Logarithm
Now we simplify each logarithm:
1. \(\log_{3}27 = \log_{3}(3^3) = 3\)
2. \(\log_{3}25 = \log_{3}(5^2) = 2\log_{3}5\)
3. \(\log_{3}49 = \log_{3}(7^2) = 2\log_{3}7\)
4. \(\log_{3}81 = \log_{3}(3^4) = 4\)
5. \(\log_{3}3 = 1\)

### Step 5: Substitute Back
Substituting these values back into the expression gives:
\[ \frac{\log_{3}5 \times 3 \times \log_{3}7}{2\log_{3}5 \times 2\log_{3}7} \times \frac{4}{1} \]

### Step 6: Cancel Out Common Terms
Now we can cancel out \(\log_{3}5\) and \(\log_{3}7\):
\[ \frac{3}{4} \times 4 = 3 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{3} \]