let `N =(log_(3) 135/log_(15) 3) - (log_(3) 5/log_(405) 3)` , then N equals

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem \( N = \left( \frac{\log_{3} 135}{\log_{15} 3} \right) - \left( \frac{\log_{3} 5}{\log_{405} 3} \right) \), we can follow these steps:

### Step 1: Rewrite the logarithms using the change of base formula
Using the change of base formula, we can rewrite \( \log_{15} 3 \) and \( \log_{405} 3 \) as follows:
\[ \log_{15} 3 = \frac{\log_{3} 3}{\log_{3} 15} = \frac{1}{\log_{3} 15} \]
\[ \log_{405} 3 = \frac{\log_{3} 3}{\log_{3} 405} = \frac{1}{\log_{3} 405} \]

### Step 2: Substitute back into the expression for \( N \)
Now substituting these into the expression for \( N \):
\[ N = \log_{3} 135 \cdot \log_{3} 15 - \log_{3} 5 \cdot \log_{3} 405 \]

### Step 3: Expand \( \log_{3} 15 \) and \( \log_{3} 405 \)
Next, we can express \( \log_{3} 15 \) and \( \log_{3} 405 \) in terms of simpler logarithms:
\[ \log_{3} 15 = \log_{3} (3 \cdot 5) = \log_{3} 3 + \log_{3} 5 = 1 + \log_{3} 5 \]
\[ \log_{3} 405 = \log_{3} (135 \cdot 3) = \log_{3} 135 + \log_{3} 3 = \log_{3} 135 + 1 \]

### Step 4: Substitute these back into \( N \)
Substituting these into \( N \):
\[ N = \log_{3} 135 \cdot (1 + \log_{3} 5) - \log_{3} 5 \cdot (\log_{3} 135 + 1) \]

### Step 5: Distribute the terms
Distributing the terms gives:
\[ N = \log_{3} 135 + \log_{3} 135 \cdot \log_{3} 5 - \log_{3} 5 \cdot \log_{3} 135 - \log_{3} 5 \]

### Step 6: Simplify the expression
Notice that \( \log_{3} 135 \cdot \log_{3} 5 \) and \( -\log_{3} 5 \cdot \log_{3} 135 \) cancel each other out:
\[ N = \log_{3} 135 - \log_{3} 5 \]

### Step 7: Use the property of logarithms
Using the property of logarithms, we can combine these:
\[ N = \log_{3} \left( \frac{135}{5} \right) \]

### Step 8: Simplify the fraction
Calculating \( \frac{135}{5} \):
\[ \frac{135}{5} = 27 \]
Thus, we have:
\[ N = \log_{3} 27 \]

### Step 9: Express \( 27 \) as a power of \( 3 \)
Since \( 27 = 3^3 \):
\[ N = \log_{3} (3^3) = 3 \cdot \log_{3} 3 \]

### Step 10: Simplify further
Since \( \log_{3} 3 = 1 \):
\[ N = 3 \]

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