Find the value of `log_(49) 16807 -log_9 27`

Options

A. 0
B. 1
C. (3/2)
D. (-1)

Answer

To solve the expression \( \log_{49} 16807 - \log_{9} 27 \), we will break it down step by step.

### Step 1: Rewrite the logarithms using properties of logarithms
We can use the change of base formula for logarithms, which states that \( \log_{a} b = \frac{\log_{c} b}{\log_{c} a} \). For simplicity, we will convert both logarithms to base 7 and base 3.

### Step 2: Simplify \( \log_{49} 16807 \)
First, we recognize that:
- \( 49 = 7^2 \)
- \( 16807 = 7^5 \)

Using the change of base formula:
\[ \log_{49} 16807 = \frac{\log_{7} 16807}{\log_{7} 49} = \frac{\log_{7} (7^5)}{\log_{7} (7^2)} = \frac{5}{2} \]

### Step 3: Simplify \( \log_{9} 27 \)
Next, we recognize that:
- \( 9 = 3^2 \)
- \( 27 = 3^3 \)

Using the change of base formula:
\[ \log_{9} 27 = \frac{\log_{3} 27}{\log_{3} 9} = \frac{\log_{3} (3^3)}{\log_{3} (3^2)} = \frac{3}{2} \]

### Step 4: Combine the results
Now we can substitute these results back into the original expression:
\[ \log_{49} 16807 - \log_{9} 27 = \frac{5}{2} - \frac{3}{2} \]

### Step 5: Perform the subtraction
\[ \frac{5}{2} - \frac{3}{2} = \frac{5 - 3}{2} = \frac{2}{2} = 1 \]

### Final Answer
Thus, the value of \( \log_{49} 16807 - \log_{9} 27 \) is \( 1 \).

---