To solve the equation $\log_3 (x^2 + 4x + 12) = 2$, follow these steps:

Step 1: Rewrite the equation in exponential form

The equation is in logarithmic form. To solve it, we can convert it into an exponential form. The general rule for this conversion is:

$\log_b (A) = C \implies A = b^C$

For this case:

$\log_3 (x^2 + 4x + 12) = 2 \implies x^2 + 4x + 12 = 3^2$

Step 2: Simplify

Now, calculate $3^2$:

$x^2 + 4x + 12 = 9$

Step 3: Rearrange the equation

Now subtract 9 from both sides:

$x^2 + 4x + 12 - 9 = 0$

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

Step 4: Factor the quadratic

We can factor the quadratic equation $x^2 + 4x + 3$:

$(x + 1)(x + 3) = 0$

Step 5: Solve for $x$

Set each factor equal to zero:

$x + 1 = 0 \quad \text{or} \quad x + 3 = 0$

$x = -1 \quad \text{or} \quad x = -3$

Step 6: Check for extraneous solutions

Since the original equation involved a logarithm, we must check that the argument of the logarithm is positive for both solutions. The argument is $x^2 + 4x + 12$. Let's check this for both solutions:

• For $x = -1$:

$x^2 + 4x + 12 = (-1)^2 + 4(-1) + 12 = 1 - 4 + 12 = 9 \quad (\text{positive, valid solution})$

• For $x = -3$:

$x^2 + 4x + 12 = (-3)^2 + 4(-3) + 12 = 9 - 12 + 12 = 9 \quad (\text{positive, valid solution})$

Both solutions are valid.

Final Answer:

The solutions are $x = -1$ and $x = -3$.

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Would you like further details or clarifications on the solution? Here are some follow-up questions that might interest you:

1. What is the graph of the function $f(x) = \log_3(x^2 + 4x + 12)$?
2. How do logarithmic and exponential functions relate to each other?
3. How can you solve logarithmic equations with different bases?
4. What are the properties of logarithmic functions?
5. How do you solve quadratic equations when factoring isn't possible?

Tip: Always check for extraneous solutions when dealing with logarithmic and exponential equations!