Solving Logarithmic Equations: A Step-by-Step Guide

Hey everyone, let's dive into solving the logarithmic equation: $\log_3(x^2 + 6x) = \log_3(2x + 12)$. This might look a bit intimidating at first, but trust me, it's totally manageable! We'll break it down step-by-step to find the value(s) of x that make this equation true. Understanding how to solve logarithmic equations is crucial in various areas of mathematics, from calculus to computer science, so let's get started!

Understanding the Basics of Logarithms

Before we jump into the problem, let's quickly recap what logarithms are all about. In simple terms, a logarithm answers the question: "To what power must we raise the base to get a certain number?" In our equation, the base is 3. So, the equation $\log_3(x^2 + 6x) = \log_3(2x + 12)$ is essentially asking, "What value of x makes the power of 3 equal to the same value on both sides?" That's why the bases are the same in the equation. A key property of logarithms states that if $\log_b(M) = \log_b(N)$, then M = N, assuming both M and N are positive. This property is our key to unlocking this problem. It's like having a secret code that simplifies things – if the logs are equal and the bases are the same, the stuff inside the logs has to be equal too! Remember, guys, the properties of logarithms are your best friends when tackling these problems. They are the keys to simplifying and solving equations.

So, if we take a look at the equation, $\log_3(x^2 + 6x) = \log_3(2x + 12)$, we can apply this rule. We get $x^2 + 6x = 2x + 12$. Now, we’re left with a quadratic equation, which we can solve using basic algebra, such as rearranging terms and factoring. Notice that we now have an easier equation to work with! The core idea is to transform the original equation into a form that's easier to handle using basic algebraic principles. This transformation is the secret sauce to successfully solving logarithmic equations and unlocking the values of x!

Solving the Quadratic Equation: Step-by-Step

Alright, now that we've simplified the equation to $x^2 + 6x = 2x + 12$, let's solve this quadratic equation. The first thing we want to do is move all the terms to one side, setting the equation equal to zero. This will allow us to use factoring or the quadratic formula to find the solutions.

Let’s subtract $2x$ and $12$ from both sides, which gives us:

$x^2 + 6x - 2x - 12 = 0$

Simplifying this, we get:

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

Now, we need to factor this quadratic equation. We are looking for two numbers that multiply to -12 and add up to 4. These numbers are 6 and -2. Therefore, we can factor the quadratic equation as follows:

$(x + 6)(x - 2) = 0$

To find the possible values of x, we set each factor equal to zero and solve for x:

• $x + 6 = 0 => x = -6$
• $x - 2 = 0 => x = 2$

So, we have two potential solutions: x = -6 and x = 2. But hey, hold on a sec! We're not quite done yet. It’s super important to remember that solutions to a logarithmic equation must always be checked to make sure they're valid. Why? Because the arguments (the stuff inside the logarithm) must be positive. This is a fundamental rule of logarithms, and missing it can lead to incorrect answers.

Checking the Solutions for Validity

Now comes the crucial part: checking our solutions to make sure they are valid. As mentioned before, the argument of a logarithm (the expression inside the parentheses) must always be positive. This is super important because logarithms are only defined for positive numbers. Let's plug each solution back into the original equation and check if it holds true.

First, let's check x = -6:

• For $\log_3(x^2 + 6x)$, we get $\log_3((-6)^2 + 6(-6)) = \log_3(36 - 36) = \log_3(0)$.

• Since the logarithm of 0 is undefined, x = -6 is not a valid solution.

Next, let's check x = 2:

• For $\log_3(x^2 + 6x)$, we get $\log_3((2)^2 + 6(2)) = \log_3(4 + 12) = \log_3(16)$.

• For $\log_3(2x + 12)$, we get $\log_3(2(2) + 12) = \log_3(4 + 12) = \log_3(16)$.

• Since both sides of the equation are defined and equal when x = 2, x = 2 is a valid solution.

So, after checking both solutions, we can confidently say that only x = 2 is the valid solution to the given logarithmic equation. This step is a must-do to ensure you get the right answer and don’t fall into any traps! Always, always check your solutions – trust me, it's worth the extra effort.

Conclusion: The Final Answer

Alright, guys, we've successfully navigated the process of solving the logarithmic equation $\log_3(x^2 + 6x) = \log_3(2x + 12)$. We’ve gone through the steps of simplifying the equation, solving the resulting quadratic equation, and, most importantly, verifying our solutions to ensure they are valid. By following these steps, we found that x = 2 is the only solution that satisfies the given equation. So, the correct answer is D. x = 2. It’s a great example of how mathematical concepts build on each other, from understanding logarithms to solving quadratic equations. The key takeaway is to break down problems into smaller, manageable steps. This not only makes the process easier but also ensures that you don't miss any critical details.

In essence, solving logarithmic equations involves transforming the logarithmic form into an easier algebraic equation, solving it, and validating your answers. Remember to always check your solutions! Practice these types of problems, and you'll become a pro in no time! Keep practicing, keep learning, and keep asking questions. You've got this!