Solving Logarithmic Equations: Find X In Log₃(-18x+126)-log₃(6)=2

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Hey guys! Today, we're diving into the exciting world of logarithmic equations! Logarithms might seem intimidating at first, but trust me, with a little practice, you'll be solving them like a pro. We're going to break down a specific problem step-by-step, so you can see exactly how it's done. Our mission, should we choose to accept it, is to find the value of x that satisfies the equation: log₃(-18x + 126) - log₃(6) = 2. So, grab your thinking caps, and let's get started!

Understanding Logarithms

Before we jump into solving, 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. Basically, it answers the question: “To what power must we raise the base to get this number?” For example, log₂8 = 3 because 2³ = 8. Here, 2 is the base, and we need to raise it to the power of 3 to get 8. Make sense? This foundational understanding is crucial for tackling more complex logarithmic equations. When you see log_b(a) = c, remember it's just another way of saying b^c = a. Understanding this relationship will help you transform and simplify logarithmic expressions.

In our equation, we’re dealing with logarithms with a base of 3. So, we’re asking ourselves, “To what power must we raise 3 to get a certain number?” Keep this in mind as we move through the steps. Recognizing the base and how it relates to the logarithm is the first step in demystifying these types of problems. Remember, practice makes perfect! The more you work with logarithms, the more intuitive they’ll become. So, let's keep going and conquer this equation together! This basic understanding allows us to manipulate equations and eventually isolate the variable we are trying to solve for, which in this case, is x. Don't worry if it feels a bit abstract now; we'll see it in action as we solve the problem.

Step 1: Condensing the Logarithms

Our equation looks a little cluttered right now, with two separate logarithm terms on the left side. The first thing we're going to do is condense these logarithms into a single, more manageable term. This is where the properties of logarithms come in handy! One of the most useful properties is the quotient rule, which states that: log_b(a) - log_b(c) = log_b(a/c). Notice that the bases of our logarithms are the same (both are base 3), so we can apply this rule directly. This is super important, guys – you can only combine logarithms using these rules if they have the same base! Applying the quotient rule to our equation, log₃(-18x + 126) - log₃(6) = 2, we get: log₃((-18x + 126) / 6) = 2. Awesome! We've successfully combined two logarithms into one. See how much cleaner that looks? This step alone makes the equation much less intimidating. Now, the equation is in a form that's easier to work with, and we're one step closer to solving for x. This condensing process is a common strategy when dealing with logarithmic equations, and mastering it will significantly improve your problem-solving skills. Remember, when subtracting logarithms with the same base, you can divide their arguments inside a single logarithm. Keep this rule in your mental toolkit; you'll use it often!

Step 2: Converting to Exponential Form

Now that we have a single logarithm, it's time to convert the equation from logarithmic form to exponential form. Remember that logarithm definition we talked about earlier? It's going to be our key here! The equation log₃((-18x + 126) / 6) = 2 is equivalent to saying 3² = (-18x + 126) / 6. Do you see how we transformed it? We took the base (3), raised it to the power of the result (2), and set it equal to the argument of the logarithm ((-18x + 126) / 6). This conversion is a game-changer because it gets rid of the logarithm altogether, leaving us with a regular algebraic equation that we can solve. This is a standard technique for handling logarithmic equations, and it's absolutely essential to understand. If you're ever stuck, remember the relationship between logarithms and exponentials – they're just two sides of the same coin! This transformation allows us to apply familiar algebraic techniques to isolate and solve for x. Converting to exponential form simplifies the equation, making it much easier to manipulate and solve. Practice this step, guys; it will become second nature before you know it!

Step 3: Simplify and Solve for x

We've transformed our logarithmic equation into an exponential one: 3² = (-18x + 126) / 6. Now, let's simplify and solve for x. First, we evaluate 3², which is 9. So, our equation becomes 9 = (-18x + 126) / 6. Next, we want to get rid of the fraction, so we multiply both sides of the equation by 6: 9 * 6 = -18x + 126. This gives us 54 = -18x + 126. Now, we need to isolate the term with x. Let's subtract 126 from both sides: 54 - 126 = -18x, which simplifies to -72 = -18x. Finally, to solve for x, we divide both sides by -18: x = -72 / -18. And there you have it! x = 4. We've successfully found the value of x that satisfies the equation. Each of these steps is a fundamental algebraic manipulation. Mastering these skills is crucial not just for solving logarithmic equations but for a wide range of mathematical problems. Remember, the goal is always to isolate the variable, and we do that by performing inverse operations on both sides of the equation.

Step 4: Checking for Extraneous Solutions

We've found a solution for x, but we're not quite done yet! With logarithmic equations, it's crucial to check for extraneous solutions. Extraneous solutions are values that satisfy the transformed equation but not the original logarithmic equation. This happens because logarithms are only defined for positive arguments. So, we need to make sure that plugging x = 4 back into the original equation doesn't result in taking the logarithm of a negative number or zero. Let's substitute x = 4 into the original equation: log₃(-18(4) + 126) - log₃(6) = 2. This simplifies to log₃(-72 + 126) - log₃(6) = 2, which further simplifies to log₃(54) - log₃(6) = 2. Now, we can use the quotient rule again: log₃(54/6) = 2, which gives us log₃(9) = 2. Is this true? Yes, because 3² = 9. So, x = 4 is a valid solution. This step is non-negotiable when dealing with logarithms. It's easy to get caught up in the algebra and forget to check, but doing so could lead you to an incorrect answer. Always plug your solution back into the original equation and verify that it works. Extraneous solutions often arise because the domain of logarithmic functions is restricted to positive numbers. By checking, we ensure that our solution falls within this domain and is, therefore, valid. You've got to be a detective and make sure your solution doesn't break any rules of logarithms!

Conclusion

Great job, guys! We've successfully solved the logarithmic equation log₃(-18x + 126) - log₃(6) = 2. We found that x = 4 is the solution, and we even checked for extraneous solutions to be extra sure. Remember, the key to solving logarithmic equations is to:

  1. Condense the logarithms using properties like the quotient rule.
  2. Convert the equation to exponential form using the definition of logarithms.
  3. Simplify and solve the resulting algebraic equation.
  4. Check for extraneous solutions by plugging your answer back into the original equation.

Logarithms might seem tricky at first, but with practice and a solid understanding of the rules, you can conquer any logarithmic equation that comes your way. Keep practicing, and you'll become a logarithm master in no time! Keep your thinking caps on, and let's tackle more math challenges together! We've seen how converting between logarithmic and exponential forms is a powerful technique. And don't forget the importance of checking your answers – that's where you truly become a math detective!