To solve the equation:

\[
\log_4(x^2 - 6x) = 2
\]

### Step 1: Rewrite the Logarithmic Equation in Exponential Form
The equation can be rewritten in exponential form as:

\[
x^2 - 6x = 4^2
\]

since \(4^2 = 16\), this becomes:

\[
x^2 - 6x = 16
\]

### Step 2: Rearrange into a Standard Quadratic Equation
Move all terms to one side:

\[
x^2 - 6x - 16 = 0
\]

### Step 3: Solve the Quadratic Equation
Now we factor the quadratic:

\[
(x - 8)(x + 2) = 0
\]

Setting each factor equal to zero:

\[
x - 8 = 0 \Rightarrow x = 8
\]
\[
x + 2 = 0 \Rightarrow x = -2
\]

### Step 4: Check for Valid Solutions
Since we are dealing with a logarithmic function, the argument \(x^2 - 6x\) must be positive.

1. For \(x = 8\):  
   \(x^2 - 6x = 8^2 - 6 \cdot 8 = 64 - 48 = 16\), which is positive.

2. For \(x = -2\):  
   \(x^2 - 6x = (-2)^2 - 6 \cdot (-2) = 4 + 12 = 16\), which is also positive.

### Solution
Both values, \( x = 8 \) and \( x = -2 \), satisfy the original equation.

\[
\boxed{x = 8 \text{ and } x = -2}
\]

Would you like further explanation on logarithmic functions, or do you have any questions?

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**Related Questions:**

1. How do you solve logarithmic equations with different bases?
2. What is the importance of checking domain constraints in logarithmic equations?
3. How do you factor a quadratic equation if it doesn't factor nicely?
4. What if the logarithmic function has a base different than an integer?
5. How do you solve exponential equations?

**Tip:** Always check the argument of a logarithmic function to ensure it stays positive for real solutions.

Step-by-step solution to solve log base 4 (x^2 - 6x) = 2 by converting to exponential form and solving the resulting quadratic equation. This example covers key concepts in logarithms and quadratic equations, ideal for high school students in Grades 9-12.

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