Solving Logarithmic Equations: Step-by-Step Solution

Hey guys! Today, we're diving into the exciting world of logarithmic equations! We're going to break down how to solve the equation log₃(x + 12) = log₃(5x) step by step. It might look a little intimidating at first, but trust me, it's totally manageable. We'll go through each step in detail, so you’ll be a pro at solving these types of problems in no time. So, grab your pencils, and let's get started!

Understanding Logarithmic Equations

Before we jump into solving our specific equation, let's quickly recap what logarithmic equations are all about. A logarithmic equation is essentially an equation where the logarithm appears. Remember, logarithms are just the inverse operation of exponentiation. So, when you see something like log base 'b' of 'a' equals 'c' (written as logь(a) = c), it's the same as saying b raised to the power of c equals a (bᶜ = a). This relationship is super important for understanding and solving logarithmic equations.

Now, when we have logarithmic equations where the bases of the logarithms are the same on both sides (like in our example, where the base is 3), we can use a cool property to simplify things. If logь(A) = logь(B), then it directly implies that A = B. This is because the logarithmic function is one-to-one, meaning that for each unique input, there's a unique output. So, if the logarithms of two expressions are equal with the same base, the expressions themselves must be equal. This property is our key to unlocking the solution to our equation, and we’ll use it in the next section to simplify things significantly. Understanding this foundational concept makes solving logarithmic equations so much easier, so let’s keep this in mind as we proceed.

Step 1: Apply the Property of Logarithms

Okay, let's get down to business! Our equation is log₃(x + 12) = log₃(5x). The first thing we notice is that we have logarithms with the same base (base 3) on both sides of the equation. This is perfect because it allows us to use that nifty property we just discussed: If logь(A) = logь(B), then A = B. In our case, 'A' is (x + 12) and 'B' is (5x). So, we can directly equate the arguments of the logarithms.

By applying this property, we transform our logarithmic equation into a much simpler algebraic equation. We can now rewrite log₃(x + 12) = log₃(5x) as x + 12 = 5x. See how much cleaner that looks? This step is crucial because it eliminates the logarithms, allowing us to work with a linear equation that we can easily solve using basic algebra. Essentially, we've peeled away a layer of complexity, bringing us closer to finding the value of 'x'. This transformation is the heart of solving many logarithmic equations, so understanding why and how we do this is super important. We’ve now set the stage for solving for 'x'; let’s move on to the next step and actually isolate 'x'.

Step 2: Solve for x

Alright, we've got our simplified equation: x + 12 = 5x. Now it's time to roll up our sleeves and solve for 'x'. Our goal is to isolate 'x' on one side of the equation. The first thing we can do is subtract 'x' from both sides. This gets all the 'x' terms on one side and the constant terms on the other, which is a classic algebraic strategy. Subtracting 'x' from both sides gives us 12 = 5x - x, which simplifies to 12 = 4x. We're getting closer!

Now, we just have one more step to isolate 'x'. We have 12 = 4x, and we want to get 'x' by itself. What's the opposite of multiplying by 4? Dividing by 4, of course! So, we divide both sides of the equation by 4. This gives us 12 / 4 = 4x / 4, which simplifies to 3 = x. So, we've found a potential solution: x = 3. But hold on, we're not quite done yet! It’s super important in logarithmic equations to check our solution, and we’ll discuss why in the next section.

Step 3: Check the Solution

Okay, we've found a potential solution: x = 3. But here's the deal with logarithmic equations: we always need to check our solution. Why? Because logarithms are only defined for positive arguments. In other words, you can't take the logarithm of a negative number or zero. If we plug our solution into the original equation and end up taking the logarithm of a negative number or zero, that solution is what we call extraneous, meaning it's not a valid solution.

So, let's plug x = 3 back into our original equation: log₃(x + 12) = log₃(5x). Substituting x = 3, we get log₃(3 + 12) = log₃(5 * 3). This simplifies to log₃(15) = log₃(15). Awesome! Both arguments, (15) and (15), are positive, so x = 3 is a valid solution. If we had plugged in our solution and found a negative number or zero inside the logarithm, we would have to discard that solution. Checking solutions is a critical step in solving logarithmic equations, and forgetting this step can lead to incorrect answers. In our case, x = 3 checks out, so we can confidently say that it is indeed the solution to our equation.

Conclusion

And there you have it! We've successfully solved the logarithmic equation log₃(x + 12) = log₃(5x). We walked through each step, from applying the fundamental property of logarithms to simplify the equation, to solving for 'x', and finally, to the crucial step of checking our solution. Remember, the key takeaway here is that logarithmic equations might seem tricky at first, but by breaking them down step by step and remembering to check your answers, you can totally conquer them.

The solution we found was x = 3, which we verified by plugging it back into the original equation. This entire process highlights the importance of understanding the properties of logarithms and the necessity of checking solutions in these types of problems. So, next time you encounter a logarithmic equation, don't sweat it! Just follow these steps, and you'll be solving them like a pro. Keep practicing, and you'll become even more confident in your ability to tackle these types of mathematical challenges. Great job, guys!