Unlocking The Exponential Equation: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the fascinating world of exponential equations. Specifically, we'll be tackling the equation $2\left(3^{y-2}\right)=18$. Don't worry if it looks a bit intimidating at first; we'll break it down step-by-step to make it super clear and easy to understand. This is a classic example of how we use logarithms and exponent rules to find the value of an unknown variable. This kind of problem pops up in a lot of different areas of math and science, so getting a handle on it is definitely worth your time. Let's get started and see how we can unravel this mathematical puzzle! The journey to solving this equation is a fun one.

Before we begin our mathematical adventure, let's take a quick look at the core concepts we'll be using. At its heart, an exponential equation is one where the variable we're trying to solve for appears as an exponent. In our case, the variable 'y' is sitting right up there in the exponent of 3. Understanding the rules of exponents and logarithms is absolutely key to solving these types of equations. When we're working with exponents, remember that the base is the number being multiplied and the exponent tells us how many times to multiply the base by itself. For example, in $3^2$, the base is 3, and the exponent is 2, so the answer is 3*3 = 9. Logarithms are the inverse of exponents, which means they "undo" exponentiation. The logarithm tells us to what power we need to raise a base to get a certain number. Don't worry if that sounds a bit confusing; we'll use these ideas as we work through the problem. Keep in mind that we're essentially trying to isolate the exponential part of the equation and then use either logarithms or a bit of algebra to solve for 'y'. This process involves using the properties of exponents and logarithms to simplify the equation and get our answer. This equation is an awesome illustration of how to blend algebra and exponential functions. The strategies we apply here are useful for a wide array of problems. So, buckle up!

One of the most useful things to keep in mind is the order of operations. We're going to use this idea a lot as we solve the exponential equation. In the process, you'll see why it's so important to solve equations with exponents. Let's make sure we're on the right path when we start solving exponential equations.

Isolating the Exponential Term: Our First Step

Alright, guys, let's jump right into it! The first thing we need to do is get that exponential term all by itself. We've got $2\left(3^{y-2}\right)=18$, and our goal is to isolate the $3^{y-2}$ part. How do we do that? Simple, we're going to get rid of that pesky '2' that's multiplying everything. We achieve this by dividing both sides of the equation by 2.

So, let's do it! Divide both sides by 2:

$\frac{2\left(3^{y-2}\right)}{2}=\frac{18}{2}$

The 2's on the left side cancel each other out, leaving us with:

$3^{y-2} = 9$

Nice! We've successfully isolated the exponential term. Now the equation looks a lot cleaner and easier to manage. This is a crucial step because it simplifies our equation to make it manageable. Remember, our main objective is to find the value of 'y', and isolating the exponential term gets us one step closer to our goal. This technique is super important for solving many kinds of exponential equations. Now we're in a position to easily proceed with the next step. Notice how isolating the exponential term helps streamline the procedure, which means it reduces the amount of work required and helps you solve it quicker. Always strive to simplify things to make your life easier.

This simple division has significantly changed the nature of our equation. The goal is to get the exponent by itself, which will help us solve the equation more easily. This concept is fundamental for your ability to solve more complex equations. By making sure we apply each step, we can solve any similar math problem. This makes math so rewarding, seeing how each step solves the equation. The exponential part of the equation has been successfully isolated.

Solving for the Exponent

Now that we've got $3^{y-2} = 9$, we need to figure out how to get to 'y'. You might already see it, but we can rewrite 9 as a power of 3. Remember that $3^2$ means 3 multiplied by itself twice, which equals 9. So, we can rewrite our equation as:

$3^{y-2} = 3^2$

When the bases are the same, the exponents must be equal. This is the key to solving this type of equation. Since the bases are both 3, we can equate the exponents:

$y - 2 = 2$

This is a simple linear equation. To solve for 'y', we just need to add 2 to both sides:

$y - 2 + 2 = 2 + 2$

This simplifies to:

$y = 4$

Boom! We've found the value of 'y'! We've successfully navigated the exponential equation and arrived at a solution. This step is about recognizing the power relationships between numbers. The important thing to keep in mind here is that when you have the same base on both sides of an equation, you can set the exponents equal to each other. This is the core principle behind this step and it's a technique that's used over and over in exponential equations. Always look for ways to relate the numbers involved to powers of the same base. You'll see that it's a game-changer. It is a powerful method. Remember that you can always check your answer by substituting 'y = 4' back into the original equation to make sure it holds true.

This step leverages our understanding of exponents and equivalent expressions. By expressing both sides of the equation with the same base, we've transformed the problem into a simpler algebraic equation. This approach is widely applicable in exponential equation solving. It's awesome to observe how we transform a tricky exponential equation into a basic linear one. The core concept here is that if the bases are the same, the exponents must also be the same. That's a fundamental principle. This is the heart of solving these types of equations.

Verification: Putting Our Solution to the Test

Always a good idea to double-check our work. Let's substitute $y = 4$ back into the original equation: $2\left(3^{y-2}\right)=18$. Replace 'y' with 4:

$2\left(3^{4-2}\right)=18$

Simplify the exponent:

$2\left(3^{2}\right)=18$

Calculate $3^2$:

$2(9) = 18$

Multiply:

$18 = 18$

Success! The equation holds true when y = 4. This confirms that our solution is correct. Verification is a critical step in math. It helps us ensure we haven't made any mistakes. It's a way to boost your confidence in your answer. This step is super important in any problem to make sure the answer is correct. If the equation checks out, then you can be confident that you've solved it correctly. Always verify! The process of checking our work helps to reinforce our understanding. We can be 100% sure we've found the correct value for 'y'!

This verification process not only confirms our solution but also reinforces the principles behind solving exponential equations. By substituting the value back into the original equation, we are essentially retracing our steps and ensuring that the logic holds true. It also gives us a chance to practice our arithmetic and algebraic skills. Always remember to check your solutions; it's a key habit to build in any mathematical endeavor. Congratulations! You've successfully solved the exponential equation.

Conclusion: Mastering Exponential Equations

Awesome work, guys! We've successfully solved the exponential equation $2\left(3^{y-2}\right)=18$, and we've learned some cool strategies along the way. Remember, the key steps are to isolate the exponential term, rewrite the equation with the same base, and then solve for the exponent. These techniques are super useful, so make sure you practice them. The skills we have practiced today will be super valuable as you go through more advanced math topics. Keep exploring, keep practicing, and don't be afraid to tackle new challenges. You're building a strong foundation in mathematics, and that's something to be proud of. Keep practicing these types of problems, and you'll find that you can solve many different kinds of exponential equations with confidence. Go forth and conquer those exponential equations!

Exponential equations are present throughout various disciplines. This approach serves as a basis for solving more complex equations. The steps outlined provide a structured method to unravel this equation. Embrace the challenge, and remember that with practice, these equations become much easier. Keep practicing. Keep exploring. Keep having fun with math! This process is a testament to the power of mathematics. You're well on your way to becoming math masters!