Solve : `log_(5-x)(x^(2)-2x+65)=2`

Answer

To solve the equation \( \log_{(5-x)}(x^2 - 2x + 65) = 2 \), we will follow these steps:

### Step 1: Rewrite the logarithmic equation in exponential form
Using the property of logarithms, we can rewrite the equation:
\[ x^2 - 2x + 65 = (5 - x)^2 \]

### Step 2: Expand the right-hand side
Now, we will expand the right-hand side:
\[ (5 - x)^2 = 25 - 10x + x^2 \]
So, we have:
\[ x^2 - 2x + 65 = 25 - 10x + x^2 \]

### Step 3: Simplify the equation
Next, we can simplify the equation by subtracting \(x^2\) from both sides:
\[ -2x + 65 = 25 - 10x \]
Now, we will rearrange the terms:
\[ -2x + 10x = 25 - 65 \]
This simplifies to:
\[ 8x = -40 \]

### Step 4: Solve for \(x\)
Now, we can solve for \(x\) by dividing both sides by 8:
\[ x = \frac{-40}{8} = -5 \]

### Step 5: Verify the solution
We should check if our solution satisfies the original logarithmic equation. The base of the logarithm, \(5 - x\), must be positive and not equal to 1. For \(x = -5\):
\[ 5 - (-5) = 10 > 0 \quad \text{(valid base)} \]
Now, we check the argument:
\[ x^2 - 2x + 65 = (-5)^2 - 2(-5) + 65 = 25 + 10 + 65 = 100 > 0 \quad \text{(valid argument)} \]
Thus, both conditions for the logarithm are satisfied.

### Final Answer
The solution to the equation is:
\[ \boxed{-5} \]