Solving Logarithmic Equations: Finding The Value Of X

Hey guys! Let's dive into a fun math problem today. We're going to solve a logarithmic equation. Specifically, we're looking at: If log base 2 of (3x+6) equals 2, what is the value of x? This might seem a little intimidating at first, but trust me, it's totally manageable. We'll break it down step by step, making sure you understand every bit of it. Logarithms are super useful in all sorts of fields, from computer science to finance, so understanding them is a great skill to have. Ready to get started?

Understanding the Basics of Logarithms

Okay, before we jump into solving the equation, let's make sure we're all on the same page about what logarithms actually are. Think of a logarithm as the inverse operation of exponentiation (raising a number to a power). When we write log base b of a = c, it's the same as saying b raised to the power of c equals a, or b^c = a. The 'b' is called the base, 'a' is the argument or the number we're taking the log of, and 'c' is the exponent or the answer to the logarithm. For example, log base 2 of 8 = 3 because 2^3 = 8. Got it? It's like asking, "To what power must we raise the base (2 in this case) to get the argument (8)?" The answer is 3. Now, in our original problem, we have log base 2 of (3x + 6) = 2. Here, the base is 2, the argument is (3x + 6), and the exponent is 2. Our job is to figure out what x has to be to make this true. We're basically working backward from the exponent to find the unknown variable hidden inside the argument. This process is all about understanding the relationship between logarithms and exponents and being able to convert between the two forms seamlessly. Mastering this is key to solving a wide variety of logarithmic problems. So, if you're ever stuck, just remember the relationship log base b of a = c is equivalent to b^c = a. This little trick will take you a long way. It's crucial to grasp the definitions because, without a solid understanding of this, we won't be able to solve for x. We're going to use this knowledge in the next section to tackle our problem directly.

The Core Concept of Logarithms

Remember, the fundamental concept is about finding the exponent. Logarithms are essentially a different way of writing exponential equations. They're about asking, "What power do I need to raise the base to in order to get a certain number?" When you see log base 2 of 8 = 3, you're really asking, "2 to the power of what equals 8?" And the answer is 3 (because 2 cubed, or 2^3, is 8). In our case, log base 2 of (3x + 6) = 2, the question transforms to "2 to the power of what equals (3x + 6)?" The answer, according to our equation, is 2. Therefore, 2^2 = (3x + 6). The core idea is to change the logarithmic expression to an exponential one, making it easier to solve the unknown variable. This is a critical first step. It transforms the problem into a simpler equation which we all know how to solve. Recognizing this conversion and working with the exponential form enables us to unravel the logarithmic equation systematically and is the heart of what we're doing. Let's make sure we are clear: Logarithms are not something to be feared. Instead, they provide a different perspective and a powerful tool in mathematical problem solving.

Solving for x Step by Step

Alright, now that we're feeling comfortable with the basics, let's get down to business and solve our equation for x. We'll follow a few simple steps. The first thing we need to do is convert the logarithmic equation to its exponential form. Remember, the equation is log base 2 of (3x + 6) = 2. This translates to 2^2 = 3x + 6. Simple, right? See, we've replaced the log with a plain old exponent! The next step is to simplify the exponential part. We know that 2^2 equals 4. So our equation becomes 4 = 3x + 6. Now we have a straightforward algebraic equation. Now, we'll isolate the 'x' term. To do this, we need to get the 'x' all by itself on one side of the equation. We start by subtracting 6 from both sides of the equation. This gives us 4 - 6 = 3x + 6 - 6, which simplifies to -2 = 3x. Now, to get 'x' completely alone, we divide both sides by 3. This gives us -2/3 = x. So, x = -2/3.

Detailed Breakdown of the Solution

Let's meticulously walk through each step to ensure clarity. We began with log base 2 of (3x + 6) = 2. Step 1: Convert to Exponential Form. 2^2 = 3x + 6. Step 2: Simplify the exponent. 4 = 3x + 6. Step 3: Isolate the 'x' term. Subtract 6 from both sides: 4 - 6 = 3x + 6 - 6, which simplifies to -2 = 3x. Step 4: Solve for 'x'. Divide both sides by 3: -2/3 = x. This means that x = -2/3. There you have it! By converting the logarithmic equation into its exponential form, and then systematically solving the resulting algebraic equation, we found the value of x. We followed a clear sequence to arrive at the solution. Each step built upon the last, making the process easily understandable. This step-by-step approach not only helps us find the right answer but also reinforces our understanding of the underlying principles. Let's quickly double-check our work. Plug -2/3 back into the original equation: log base 2 of (3*(-2/3) + 6) = 2. Simplifying the inside, 3*(-2/3) equals -2. Then, -2 + 6 = 4. So the equation is log base 2 of 4 = 2. Since 2^2 = 4, our answer is correct. Remember, verifying your answer is always a good practice. It solidifies your grasp of the material and helps prevent mistakes.

Verifying the Solution

It's always a good idea to check your answer! To make sure we've done everything correctly, let's substitute our value of x, which is -2/3, back into the original equation. Our original equation was log base 2 of (3x + 6) = 2. Replacing x with -2/3, we get log base 2 of (3*(-2/3) + 6). Let's simplify this. 3 * (-2/3) is equal to -2. So, we now have log base 2 of (-2 + 6). Further simplifying, -2 + 6 equals 4. Therefore, our equation becomes log base 2 of 4 = 2. Does this hold true? Well, 2^2 is indeed equal to 4. Therefore, our solution is correct! This confirmation step is super important. It gives us confidence that we've accurately applied our understanding of logarithms and algebraic manipulation. It also helps catch any silly mistakes we might have made along the way. Always take the time to check your answers; it's a fundamental part of doing math well. This habit builds accuracy and solidifies your understanding of the concepts involved.

The Importance of Verification

Verifying our answer is more than just a formality; it is an important step. By substituting our solution back into the original equation, we gain confidence in our method and understanding. When we insert x = -2/3 into our equation, we end up simplifying to log base 2 of 4 = 2. This step shows that if the value of x, -2/3, is correct, the equation should balance. The right-hand side of the equation should equal the left side of the equation. Since 2^2 equals 4, the logarithmic expression becomes true. The verification step offers us a method to validate the accuracy of our solutions. This step reduces the chance of errors. Furthermore, this also helps to strengthen our problem-solving skills and boosts our understanding of logarithms and exponential functions. Verification is crucial to improve our grasp of mathematical concepts and build confidence in our abilities. So, always remember to verify your work! It is a crucial step in ensuring accuracy and improving your understanding of mathematical principles.

Conclusion and Next Steps

Awesome work, guys! We've successfully solved the logarithmic equation and found that x = -2/3. We started with log base 2 of (3x + 6) = 2 and, through the power of converting to exponential form and some simple algebra, we got our answer. We also verified our solution, which is always a smart thing to do. Remember, the key to solving logarithmic equations is understanding the relationship between logarithms and exponents and being able to convert between the two. Keep practicing! The more you work through these types of problems, the easier and more intuitive they will become. Now, you should be able to solve similar logarithmic problems with confidence. The more you solve different types of equations, the more familiar you will become with the methods and techniques. If you want to dive deeper, you can try some additional practice problems or explore more complex logarithmic equations, such as those that involve the change of base formula or different types of logarithms (like the natural logarithm, which uses the base 'e').

Practicing Logarithmic Problems

To solidify your understanding, try working through more logarithmic problems on your own. Practice is essential! Start with simple exercises. Then, gradually work your way up to problems that incorporate more complex expressions. By consistently practicing, you will become more comfortable with these equations. Try a variety of exercises that test different concepts. This will help strengthen your skills and allow you to understand how to approach logarithmic equations. Make sure to work with the different forms of these equations. This will help improve your understanding of the different forms. If you're looking for more exercises, you can find a lot of great resources online or in textbooks. The more you do, the better you'll become at solving these types of problems. Remember, the goal is not just to get the right answer, but to understand the