The value of `(log_(3)54)/(log_(486)3) - (log_(3)1458)/(log_(18)3)` is:

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the expression \(\frac{\log_{3}54}{\log_{486}3} - \frac{\log_{3}1458}{\log_{18}3}\), we can use the change of base formula for logarithms, which states that \(\log_{b}a = \frac{\log_{k}a}{\log_{k}b}\) for any positive \(k\).

### Step 1: Apply the Change of Base Formula
Using the change of base formula, we can rewrite the logarithms in the denominators:

\[ \log_{486}3 = \frac{\log_{3}3}{\log_{3}486} = \frac{1}{\log_{3}486} \]
\[ \log_{18}3 = \frac{\log_{3}3}{\log_{3}18} = \frac{1}{\log_{3}18} \]

### Step 2: Substitute Back into the Expression
Now substitute these back into the original expression:

\[ \frac{\log_{3}54}{\frac{1}{\log_{3}486}} - \frac{\log_{3}1458}{\frac{1}{\log_{3}18}} = \log_{3}54 \cdot \log_{3}486 - \log_{3}1458 \cdot \log_{3}18 \]

### Step 3: Simplify Each Term
Next, we can simplify each logarithm using the properties of logarithms.

1. **For \(\log_{3}54\)**:
\[ 54 = 2 \cdot 3^3 \implies \log_{3}54 = \log_{3}(2) + 3 \]

2. **For \(\log_{3}1458\)**:
\[ 1458 = 2 \cdot 3^6 \implies \log_{3}1458 = \log_{3}(2) + 6 \]

3. **For \(\log_{3}486\)**:
\[ 486 = 2 \cdot 3^5 \implies \log_{3}486 = \log_{3}(2) + 5 \]

4. **For \(\log_{3}18\)**:
\[ 18 = 2 \cdot 3^2 \implies \log_{3}18 = \log_{3}(2) + 2 \]

### Step 4: Substitute Simplified Values
Now substitute these values back into the expression:

\[ (\log_{3}(2) + 3)(\log_{3}(2) + 5) - (\log_{3}(2) + 6)(\log_{3}(2) + 2) \]

### Step 5: Expand Both Products
Now we expand both products:

1. **First Product**:
\[ (\log_{3}(2) + 3)(\log_{3}(2) + 5) = \log_{3}(2)^2 + 8\log_{3}(2) + 15 \]

2. **Second Product**:
\[ (\log_{3}(2) + 6)(\log_{3}(2) + 2) = \log_{3}(2)^2 + 8\log_{3}(2) + 12 \]

### Step 6: Combine the Results
Now combine the results:

\[ (\log_{3}(2)^2 + 8\log_{3}(2) + 15) - (\log_{3}(2)^2 + 8\log_{3}(2) + 12) = 15 - 12 = 3 \]

### Final Answer
Thus, the value of the expression is:

\[ \boxed{3} \]