The value of `log_((sqrt(2)-1))(5sqrt(2)-7)` is :

Options

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

Answer

To find the value of \( \log_{(\sqrt{2}-1)}(5\sqrt{2}-7) \), we can follow these steps:

### Step 1: Simplify \( 5\sqrt{2}-7 \)
We start by rewriting \( 5\sqrt{2}-7 \) in a form that can be related to \( \sqrt{2}-1 \).

### Step 2: Square \( \sqrt{2}-1 \)
Calculating \( (\sqrt{2}-1)^2 \):
\[ (\sqrt{2}-1)^2 = 2 - 2\sqrt{2} + 1 = 3 - 2\sqrt{2} \]

### Step 3: Multiply \( 3-2\sqrt{2} \) by \( \sqrt{2}-1 \)
Next, we multiply \( 3-2\sqrt{2} \) by \( \sqrt{2}-1 \):
\[ (3 - 2\sqrt{2})(\sqrt{2}-1) = 3\sqrt{2} - 3 - 2\sqrt{2}\cdot\sqrt{2} + 2\sqrt{2} = 3\sqrt{2} - 3 - 4 + 2\sqrt{2} \]
\[ = 5\sqrt{2} - 7 \]

### Step 4: Relate \( 5\sqrt{2}-7 \) to \( \sqrt{2}-1 \)
From the calculation in Step 3, we have shown that:
\[ 5\sqrt{2} - 7 = (\sqrt{2} - 1)^3 \]

### Step 5: Rewrite the logarithm
Now we can rewrite our logarithmic expression:
\[ \log_{(\sqrt{2}-1)}(5\sqrt{2}-7) = \log_{(\sqrt{2}-1)}((\sqrt{2}-1)^3) \]

### Step 6: Apply the logarithm power rule
Using the logarithm power rule \( \log_b(a^n) = n \cdot \log_b(a) \):
\[ \log_{(\sqrt{2}-1)}((\sqrt{2}-1)^3) = 3 \cdot \log_{(\sqrt{2}-1)}(\sqrt{2}-1) \]

### Step 7: Evaluate the logarithm
Since \( \log_{(\sqrt{2}-1)}(\sqrt{2}-1) = 1 \):
\[ 3 \cdot 1 = 3 \]

### Final Answer
Thus, the value of \( \log_{(\sqrt{2}-1)}(5\sqrt{2}-7) \) is:
\[ \boxed{3} \]