Find the minimum value of the function $$ f(x) = \log_2(x^2 - 2x + 5) $$.

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To find the minimum value of $$ f(x) = \log_2(x^2 - 2x + 5) $$, follow these steps:

**Step 1: Analyze the quadratic expression $$ x^2 - 2x + 5 $$**  
The quadratic expression represents a parabola that opens upwards since the coefficient of $$ x^2 $$ is positive. The minimum value occurs at the vertex, which is given by:  
$$
x = -\frac{b}{2a}
$$  
Here, $$ a = 1 $$ and $$ b = -2 $$:  
$$
x = \frac{-(-2)}{2(1)} = \frac{2}{2} = 1
$$

At $$ x = 1 $$, substitute into the quadratic expression to find its minimum value:  
$$
x^2 - 2x + 5 = (1)^2 - 2(1) + 5 = 1 - 2 + 5 = 4
$$

Thus, the minimum value of $$ x^2 - 2x + 5 $$ is $$ 4 $$.

**Step 2: Evaluate the logarithmic function at this point**  
Substitute the minimum value of the quadratic expression into the logarithmic function:  
$$
f(x) = \log_2(4)
$$  
Since $$ 4 = 2^2 $$, we know:  
$$
\log_2(4) = \log_2(2^2) = 2
$$

Thus, the minimum value of the function is **$$ 2 $$**.

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