We are given the equation:

$\log_3(x^2 - 4x + 4) = 1 + \log_3(x - 2)$

Step 1: Simplify the equation

We can rewrite the equation using the logarithmic properties.

The right-hand side is:

$1 + \log_3(x - 2)$

We can express 1 as $\log_3(3)$, so the right-hand side becomes:

$\log_3(3) + \log_3(x - 2)$

Using the logarithmic property $\log_b(a) + \log_b(c) = \log_b(ac)$, the right-hand side simplifies to:

$\log_3(3(x - 2))$

Thus, the equation becomes:

$\log_3(x^2 - 4x + 4) = \log_3(3(x - 2))$

Step 2: Remove the logarithms

Since the logarithms on both sides have the same base (base 3), we can remove the $\log_3$ terms and set the arguments equal to each other:

$x^2 - 4x + 4 = 3(x - 2)$

Step 3: Solve the quadratic equation

Now expand and simplify the equation:

$x^2 - 4x + 4 = 3x - 6$

Bring all terms to one side:

$x^2 - 4x + 4 - 3x + 6 = 0$

Simplify:

$x^2 - 7x + 10 = 0$

Step 4: Factor the quadratic equation

We factor the quadratic equation:

$x^2 - 7x + 10 = (x - 5)(x - 2) = 0$

Step 5: Solve for $x$

From the factored equation, the solutions are:

$x - 5 = 0 \quad \text{or} \quad x - 2 = 0$

Thus, $x = 5$ or $x = 2$.

Step 6: Check for extraneous solutions

We need to check whether any of the solutions make the argument of the logarithm negative or zero, as logarithms are undefined for such values.

• For $x = 5$:

  • $\log_3(5^2 - 4 \cdot 5 + 4) = \log_3(9)$, which is defined.
  • $\log_3(5 - 2) = \log_3(3)$, which is defined.

• For $x = 2$:

  • $\log_3(2^2 - 4 \cdot 2 + 4) = \log_3(0)$, which is undefined.

Thus, $x = 2$ is an extraneous solution.

Final Answer:

The only valid solution is:

$x = 5$

Would you like more details or clarifications on any part of the solution?

Here are 5 related questions to deepen your understanding:

1. What are the properties of logarithms that were used in this problem?
2. How can you solve logarithmic equations with different bases?
3. How do you determine if a solution is extraneous in logarithmic equations?
4. Can the quadratic formula be used to solve all quadratic equations?
5. How do logarithmic and exponential functions relate to each other?

Tip: Always check for extraneous solutions when solving logarithmic equations, as they can arise when the argument of the logarithm becomes zero or negative.