Hey guys! Today, we're diving into a cool logarithmic equation that might look a little intimidating at first glance, but trust me, it's totally doable. We're going to find the value of x in the equation 2log(3x) = 2log(121). This isn't just about crunching numbers; it's about understanding how logarithms work and how we can use their properties to isolate our unknown variable. So, grab your calculators (or just your thinking caps!), and let's break this down step-by-step. We'll explore the properties of logarithms, simplify the equation, and arrive at the solution for x. Get ready to boost your math skills!

Understanding Logarithm Properties

Before we jump into solving for x, it's super important to get comfortable with some basic logarithm properties. These are the tools in our toolbox that will help us simplify and manipulate the equation. The property we'll be using most here is the Power Rule of Logarithms, which states that log(a^b) = b*log(a). While we might not explicitly use it in the form of raising to a power, the inverse idea is also true: if you have a coefficient in front of a logarithm, you can move it to become an exponent of the argument inside the logarithm. So, b*log(a) = log(a^b). Another key property is the One-to-One Property of Logarithms. This one is a real game-changer: if log(a) = log(b), then it must be true that a = b. This is because logarithmic functions are one-to-one, meaning each output corresponds to a unique input. Think of it like this: if two different inputs gave you the same output for a logarithm, it wouldn't be a true function! So, if the logarithms on both sides of an equation have the same base (and in this case, they do, as the base is implied to be the same on both sides), we can simply equate the arguments. We'll also be dealing with the concept of the argument of a logarithm. Remember, the argument (the part inside the parentheses, like '3x' or '121' in our problem) must always be positive. This is a crucial constraint that we need to keep in mind, especially when we're solving for x, to ensure our solution is valid.

Simplifying the Equation

Alright, let's get down to business with our equation: 2log(3x) = 2log(121). The first thing you'll notice is that both sides of the equation have a factor of 2. This is fantastic because we can simplify things right away! If we divide both sides of the equation by 2, we get: (2log(3x))/2 = (2log(121))/2. This leaves us with log(3x) = log(121). See? Already looking much cleaner! Now, this is where our One-to-One Property of Logarithms comes into play. Since the logarithm on the left side is equal to the logarithm on the right side, and we assume they have the same base (usually base 10 or base e if not specified, but the base doesn't matter for this step), we can confidently state that their arguments must be equal. Therefore, we can set the expressions inside the logarithms equal to each other: 3x = 121. This step is critical because it transforms our logarithmic equation into a simple algebraic equation that we can solve for x directly. It's like unlocking the hidden value of x by stripping away the logarithmic layers. Remember, this simplification relies on the fundamental nature of logarithms as one-to-one functions. If log(A) equals log(B), then A must equal B. It's a direct consequence of how logarithms are defined and behave. So, by taking this step, we've effectively bypassed the need to deal with the logarithms themselves and are now focused solely on the algebraic relationship between the arguments.

Solving for x

We've reached the final stretch, guys! We simplified our equation down to 3x = 121. Now, our mission is to isolate x. To do this, we need to get rid of the coefficient '3' that's multiplying x. The way to undo multiplication is division. So, we'll divide both sides of the equation by 3: (3x)/3 = 121/3. This gives us x = 121/3. And there you have it! The value of x is 121/3. It's that simple! Now, before we declare victory, let's do a quick check to make sure our solution is valid. Remember that the argument of a logarithm must be positive? In our original equation, the arguments were 3x and 121. We already know 121 is positive. For 3x, if x = 121/3, then 3x = 3 * (121/3) = 121. Since 121 is positive, our solution x = 121/3 is indeed valid. This check is a really important habit to get into when solving logarithmic equations, as sometimes you might get extraneous solutions that don't satisfy the domain requirements of the logarithms. So, in summary, we used the property of logarithms to simplify the equation, turned it into a basic algebraic equation, and solved for x. The process was straightforward because of the properties we applied. It highlights how understanding these fundamental rules can make complex-looking problems much more manageable. The beauty of this problem lies in its direct application of these core logarithmic principles, leading us efficiently to the correct answer.

Final Answer and Verification

So, to recap, by applying the properties of logarithms, specifically the one-to-one property after simplifying the equation, we arrived at x = 121/3. Let's quickly verify this. If we plug x = 121/3 back into the original equation, 2log(3x) = 2log(121), we get 2log(3 * (121/3)) = 2log(121). This simplifies to 2log(121) = 2log(121), which is a true statement. This confirms that our solution is correct. It's always a good practice to perform this verification, especially in more complex problems, to catch any potential errors or extraneous solutions. The fact that the arguments of the logarithms (3x and 121) are positive for our solution further solidifies its validity. This exercise demonstrates the power and elegance of logarithmic manipulation. By leveraging fundamental properties, we can transform equations and solve for unknowns efficiently. This problem was a great example of how seemingly complex math can be broken down into manageable steps using the right tools and understanding. Keep practicing these techniques, and you'll be a logarithm pro in no time! Remember, math is all about building blocks, and each concept you master makes the next one easier to tackle. So, celebrate this win and get ready for the next challenge!