The value of `log_((8-3sqrt7))(8+3sqrt7)` is

Options

A. -2
B. -1
C. 0
D. Not defined

Answer

To solve the problem of finding the value of \( \log_{(8 - 3\sqrt{7})}(8 + 3\sqrt{7}) \), we can follow these steps:

### Step-by-step Solution:

1. **Identify the logarithmic expression**: We need to evaluate \( \log_{(8 - 3\sqrt{7})}(8 + 3\sqrt{7}) \).

2. **Rationalize the argument**: To simplify the expression, we multiply and divide \( 8 + 3\sqrt{7} \) by \( 8 - 3\sqrt{7} \):
\[ \log_{(8 - 3\sqrt{7})}(8 + 3\sqrt{7}) = \log_{(8 - 3\sqrt{7})} \left( \frac{(8 + 3\sqrt{7})(8 - 3\sqrt{7})}{(8 - 3\sqrt{7})} \right) \]

3. **Simplify the numerator**: The numerator simplifies using the difference of squares:
\[ (8 + 3\sqrt{7})(8 - 3\sqrt{7}) = 8^2 - (3\sqrt{7})^2 = 64 - 63 = 1 \]
Thus, we have:
\[ \log_{(8 - 3\sqrt{7})} \left( \frac{1}{(8 - 3\sqrt{7})} \right) \]

4. **Apply the logarithm property**: Using the property of logarithms \( \log_b \left( \frac{1}{a} \right) = -\log_b(a) \):
\[ \log_{(8 - 3\sqrt{7})} \left( \frac{1}{(8 - 3\sqrt{7})} \right) = -\log_{(8 - 3\sqrt{7})}(8 - 3\sqrt{7}) \]

5. **Evaluate the logarithm**: Since \( \log_a(a) = 1 \):
\[ -\log_{(8 - 3\sqrt{7})}(8 - 3\sqrt{7}) = -1 \]

6. **Final answer**: Therefore, the value of \( \log_{(8 - 3\sqrt{7})}(8 + 3\sqrt{7}) \) is:
\[ \boxed{-1} \]