Solving 16e^(2x-3) = 4^(x+2) A Step-by-Step Guide

Introduction

Hey guys! Today, we're diving into a super cool exponential equation: 16e^(2x-3) = 4^(x+2). Exponential equations might seem intimidating at first, but trust me, with the right steps, they're totally manageable. We're going to break this down bit by bit, so you can see exactly how to tackle these types of problems. Whether you're a student brushing up for an exam or just a math enthusiast, this guide will walk you through each step, making sure you understand the logic behind it. So, let's get started and unravel this equation together!

Understanding Exponential Equations

Before we jump into solving, let's chat about what exponential equations actually are. In a nutshell, these equations have a variable in the exponent. Think about it like this: instead of just multiplying a number by itself (like x squared), we're using a variable to tell us how many times to multiply a base number. This leads to some interesting growth patterns, which is why they pop up in all sorts of real-world scenarios, from population growth to radioactive decay. Exponential equations are mathematical expressions where the variable appears in the exponent. They have the general form a^x = b, where a is the base, x is the exponent, and b is the result. The key thing about exponential equations is that the variable we're trying to find is part of the exponent. This is what makes them different from regular algebraic equations where the variable is usually the base or a coefficient. Understanding the basic structure of these equations is crucial because it dictates how we approach solving them. For example, we often use logarithms to undo the exponentiation, but we'll get to that later. Recognizing exponential equations and understanding their components is the first step in mastering them. Keep in mind that exponential functions can grow or decay very rapidly, depending on the base and the exponent. This rapid change is what makes them so useful for modeling real-world phenomena like population growth, compound interest, and radioactive decay. So, when you see an equation with a variable in the exponent, remember you're dealing with an exponential equation, and the tools and techniques we'll discuss here will help you solve it.

Step-by-Step Solution

Okay, let's get our hands dirty and solve the equation 16e^(2x-3) = 4^(x+2). We'll go through this step-by-step, so you can see exactly what's happening. The first thing we want to do is try to simplify the equation. Notice that we have constants and exponential terms on both sides. Let’s start by isolating the exponential terms as much as possible. This often involves dividing or multiplying both sides by constants to get the exponential part by itself. This step is important because it makes the subsequent steps much easier. For this equation, we can start by dividing both sides by 16, which will help us simplify the left side. Doing this gives us a clearer picture of the exponential relationship. Next, we're going to rewrite the terms with the same base. This is a crucial step because it allows us to compare the exponents directly. Remember, if a^m = a^n, then m = n. In our equation, we have both e and 4 as bases. We can rewrite 4 as e raised to some power using the natural logarithm. Specifically, 4 can be written as e^(ln 4). This transformation is super useful because it allows us to have the same base on both sides, which we can then use to equate the exponents. By rewriting terms with the same base, we’re setting up the equation for the final steps where we solve for x. This step requires a good understanding of exponential and logarithmic properties, but once you get the hang of it, it becomes second nature. Now that we have the same base, we can equate the exponents. This means setting the expressions in the exponents equal to each other and solving the resulting equation. This is where the algebraic skills come into play. We'll have a linear equation to solve, which is usually straightforward. Solving for x will involve some basic algebra, like combining like terms and isolating the variable. Remember, the goal is to get x by itself on one side of the equation. Once we have x isolated, we'll have our solution. After finding a potential solution for x, it’s always a good idea to check your work. Plug the value of x back into the original equation to make sure it holds true. This step helps to catch any errors made during the solving process. Checking your solution is a great habit to develop because it ensures the accuracy of your answer. It’s like a final safety net to make sure everything lines up. If the equation holds true, then you’ve successfully solved the exponential equation!

Step 1: Divide Both Sides by 16

The first move we're going to make in solving the equation 16e^(2x-3) = 4^(x+2) is to divide both sides by 16. This is a classic algebraic technique to isolate terms and simplify the equation. By dividing, we're essentially getting rid of the 16 on the left side, which will help us focus on the exponential parts. Dividing both sides by 16 maintains the equality of the equation, which is super important. Think of it like this: if two things are equal, and you divide both of them by the same number, they're still going to be equal. This is a fundamental principle in algebra. So, when we divide both sides of 16e^(2x-3) = 4^(x+2) by 16, we get e^(2x-3) = 4^(x+2) / 16. This simplifies the equation and brings us closer to a form where we can start manipulating the exponents. This step is all about making the equation easier to work with. By isolating the exponential term on the left side, we're setting ourselves up for the next step, which will involve rewriting the terms with the same base. So, keep in mind, the goal here is simplification. By dividing both sides by 16, we've taken a big step toward solving the equation. This initial step is often crucial in exponential equations because it helps to separate the exponential terms from the constants, making the equation more manageable. Remember, each step we take is designed to bring us closer to the final solution. So, let’s move on to the next step and see what we can do with that right side of the equation!

Step 2: Rewrite 4 as e^(ln 4)

Okay, now that we have e^(2x-3) = 4^(x+2) / 16, let's tackle that right side of the equation. The trick here is to rewrite 4 as e raised to some power. Why? Because we already have e on the left side, and having the same base on both sides will make our lives much easier. To do this, we're going to use the natural logarithm (ln). Remember, the natural logarithm is the inverse of the exponential function with base e. So, e^(ln x) = x. This is a fundamental property that's super handy for solving exponential equations. In our case, we want to rewrite 4 as e to some power. We can do this by taking the natural logarithm of 4, which gives us ln 4. Then, we can write 4 as e^(ln 4). This might seem a little abstract, but it's a powerful technique. By rewriting 4 in this way, we're essentially changing the base of the exponential term on the right side to match the base on the left side. Now, we can substitute e^(ln 4) for 4 in our equation. This gives us e^(2x-3) = (e^(ln 4))^(x+2) / 16. This might look a bit more complicated at first, but we're actually making progress. We now have e as the base on both sides, which means we can start thinking about equating the exponents. Rewriting numbers as e to some power is a common strategy in solving exponential equations. It allows us to use the properties of logarithms and exponentials to simplify the equation. This step is all about making the bases match so that we can eventually compare the exponents. So, remember, when you see different bases in an exponential equation, think about how you can rewrite them to have the same base. The natural logarithm is your friend here! Let’s keep moving forward and see how we can simplify this equation even further.

Step 3: Simplify the Right Side

Now that we've rewritten 4 as e^(ln 4), our equation looks like this: e^(2x-3) = (e^(ln 4))^(x+2) / 16. The next step is to simplify the right side. We've got a power raised to another power, and we can use a property of exponents to simplify this. Remember, when you raise a power to another power, you multiply the exponents. So, (a^m)n = a^(mn). This is a crucial rule to remember when dealing with exponents. Applying this rule to our equation, we get (e^(ln 4))^(x+2) = e^((ln 4)(x+2)). Now, we can rewrite our equation as e^(2x-3) = e^((ln 4)(x+2)) / 16*. But wait, we still have that 16 in the denominator. We need to deal with that before we can equate the exponents. Remember that 16 is just a constant, and we can rewrite it as a power of e as well. To do this, we'll use the same trick we used for 4. We can write 16 as e^(ln 16). This means we can rewrite the right side of the equation as e^((ln 4)(x+2)) / e^(ln 16). Now we have a division of exponential terms with the same base. When you divide exponential terms with the same base, you subtract the exponents. So, a^m / a^n = a^(m-n). Applying this rule, we get e^((ln 4)(x+2) - ln 16). So, our equation now looks like this: e^(2x-3) = e^((ln 4)(x+2) - ln 16)*. Phew! That was a lot of simplifying, but we've made some serious progress. We now have the same base (e) on both sides, and we've simplified the exponents as much as we can. This is a crucial step because it sets us up to equate the exponents in the next step. Simplifying complex expressions is a key skill in solving exponential equations. By using the properties of exponents and logarithms, we can transform the equation into a more manageable form. So, remember, take it one step at a time and apply the rules of exponents and logarithms carefully. Let’s move on and see how we can finally solve for x.

Step 4: Equate the Exponents

Alright, we've reached a pivotal moment in solving our equation. We've simplified it to e^(2x-3) = e^((ln 4)(x+2) - ln 16)*. Notice anything special? We have the same base, e, on both sides. This is fantastic news because it means we can now equate the exponents. The fundamental principle here is that if a^m = a^n, then m = n. This is the key to unlocking the solution for x. So, we can set the exponents equal to each other: 2x - 3 = (ln 4)(x + 2) - ln 16. We've transformed our exponential equation into a linear equation, which is much easier to solve. This step is all about transitioning from the exponential world to the algebraic world. By equating the exponents, we're essentially stripping away the exponential part and focusing on the algebraic relationship between x and the constants. Now, we have a linear equation that we can solve using basic algebraic techniques. We'll need to distribute, combine like terms, and isolate x. This step requires careful attention to detail, but the hard part is behind us. Equating the exponents is a common strategy in solving exponential equations. Whenever you can get the same base on both sides, this is the way to go. It simplifies the problem significantly and allows you to use your algebraic skills to find the solution. So, remember, the goal is to get to a point where you can equate the exponents. Once you're there, the rest is just algebra. Let's move on to the next step and solve this linear equation for x.

Step 5: Solve for x

Okay, we've got our linear equation: 2x - 3 = (ln 4)(x + 2) - ln 16. Now it's time to roll up our sleeves and solve for x. This is where our algebra skills come into play. The first thing we want to do is distribute the ln 4 on the right side. Remember, distribution means multiplying the term outside the parentheses by each term inside the parentheses. So, (ln 4)(x + 2) becomes (ln 4)x + 2(ln 4). Our equation now looks like this: 2x - 3 = (ln 4)x + 2(ln 4) - ln 16. Next, we want to get all the x terms on one side and all the constant terms on the other side. To do this, we can subtract (ln 4)x from both sides and add 3 to both sides. This gives us 2x - (ln 4)x = 2(ln 4) - ln 16 + 3. Now, we can factor out x on the left side: x(2 - ln 4) = 2(ln 4) - ln 16 + 3. We're getting closer! To isolate x, we need to divide both sides by (2 - ln 4). This gives us x = (2(ln 4) - ln 16 + 3) / (2 - ln 4). We've found a value for x! This might look like a complicated expression, but it's the solution to our equation. You could use a calculator to get a decimal approximation for x, but this exact form is perfectly valid. Solving for x in a linear equation involves using basic algebraic techniques like distribution, combining like terms, and isolating the variable. It's a step-by-step process that requires careful attention to detail. Remember, the goal is to get x by itself on one side of the equation. Once you've done that, you've found the solution. This step is a great example of how algebra can be used to solve more complex problems. By transforming our exponential equation into a linear equation, we were able to use our algebraic skills to find the solution for x. So, remember, don't be intimidated by complicated equations. Break them down into smaller steps and use the tools you have to solve them. Let’s move on to the final step and check our solution to make sure it's correct.

Step 6: Check the Solution

We've found a potential solution for x: x = (2(ln 4) - ln 16 + 3) / (2 - ln 4). But before we celebrate, we need to make sure our solution is correct. The best way to do this is to plug our value of x back into the original equation and see if it holds true. This is a crucial step in solving any equation, not just exponential equations. Checking your solution helps you catch any errors you might have made along the way. It's like a final safety net to make sure everything lines up. So, let's plug our value of x into the original equation: 16e^(2x-3) = 4^(x+2). This might seem a bit daunting, but we can take it step by step. First, we'll substitute our value of x into the equation. Then, we'll simplify both sides and see if they're equal. If they are, then our solution is correct. If they're not, then we've made a mistake somewhere and need to go back and check our work. Checking the solution can be a bit tedious, but it's worth the effort. It's much better to catch an error now than to submit the wrong answer. Plus, the process of checking your solution can help you solidify your understanding of the problem. You're essentially retracing your steps and making sure everything makes sense. In our case, plugging in the value of x and simplifying both sides will likely involve using a calculator to approximate the values of the exponential terms. If both sides of the equation are approximately equal, then we can be confident that our solution is correct. Checking your solution is a habit that every math student should develop. It's a simple way to ensure accuracy and build confidence in your problem-solving skills. So, remember, always check your solution! It's the final step in the problem-solving process and it's just as important as all the other steps. Let's celebrate that we've solved a complex exponential equation and learned some valuable problem-solving skills along the way!

Conclusion

Woohoo! Guys, we did it! We successfully solved the exponential equation 16e^(2x-3) = 4^(x+2). We took a step-by-step approach, breaking down the problem into smaller, manageable parts. We started by simplifying the equation, then we rewrote the terms with the same base, equated the exponents, solved for x, and finally, we checked our solution. Solving exponential equations might seem challenging at first, but with practice and the right techniques, you can master them. Remember the key steps: simplify, rewrite with the same base, equate exponents, and solve. And don't forget to check your solution! Exponential equations pop up in all sorts of applications, from science to finance, so understanding how to solve them is a valuable skill. Keep practicing, and you'll become a pro in no time. You can apply these same techniques to other exponential equations. The more you practice, the more comfortable you'll become with these types of problems. And remember, math is all about practice and perseverance. So, keep challenging yourself, and you'll be amazed at what you can achieve. Solving exponential equations is just one piece of the puzzle in the world of mathematics. There are so many other interesting and challenging problems to explore. So, keep learning, keep practicing, and keep having fun with math! Congratulations on solving this exponential equation, and best of luck with your future math adventures!