Find the value of the following
`log_((5+2sqrt6)) (5-2sqrt6)`

Answer

To find the value of \( \log_{(5 + 2\sqrt{6})}(5 - 2\sqrt{6}) \), we can follow these steps:

### Step 1: Rewrite the logarithm
We start with the expression:
\[ \log_{(5 + 2\sqrt{6})}(5 - 2\sqrt{6}) \]

### Step 2: Multiply numerator and denominator
To simplify \( 5 - 2\sqrt{6} \), we can multiply it by \( \frac{(5 + 2\sqrt{6})}{(5 + 2\sqrt{6})} \):
\[ 5 - 2\sqrt{6} = \frac{(5 - 2\sqrt{6})(5 + 2\sqrt{6})}{(5 + 2\sqrt{6})} \]

### Step 3: Apply the difference of squares
Using the difference of squares formula \( a^2 - b^2 \):
\[ (5 - 2\sqrt{6})(5 + 2\sqrt{6}) = 5^2 - (2\sqrt{6})^2 = 25 - 24 = 1 \]
Thus, we have:
\[ 5 - 2\sqrt{6} = \frac{1}{5 + 2\sqrt{6}} \]

### Step 4: Substitute back into the logarithm
Now we can substitute this back into our logarithm:
\[ \log_{(5 + 2\sqrt{6})}(5 - 2\sqrt{6}) = \log_{(5 + 2\sqrt{6})}\left(\frac{1}{5 + 2\sqrt{6}}\right) \]

### Step 5: Use logarithmic properties
Using the property of logarithms that states \( \log_b\left(\frac{1}{a}\right) = -\log_b(a) \):
\[ \log_{(5 + 2\sqrt{6})}\left(\frac{1}{5 + 2\sqrt{6}}\right) = -\log_{(5 + 2\sqrt{6})}(5 + 2\sqrt{6}) \]

### Step 6: Simplify further
Since \( \log_{(5 + 2\sqrt{6})}(5 + 2\sqrt{6}) = 1 \) (because any logarithm of a number to its own base is 1), we have:
\[ -\log_{(5 + 2\sqrt{6})}(5 + 2\sqrt{6}) = -1 \]

### Final Answer
Thus, the value of \( \log_{(5 + 2\sqrt{6})}(5 - 2\sqrt{6}) \) is:
\[ \boxed{-1} \]