Solving Exponential Equations: A Step-by-Step Guide

Hey everyone! Today, we're diving into the world of exponential equations. Specifically, we'll tackle the equation $9^{x-4} = 3^{x+3}$. Don't worry if this looks intimidating at first. We'll break it down step by step, making sure it's super easy to understand. Exponential equations are a fundamental concept in algebra, popping up in all sorts of real-world scenarios, like calculating compound interest or modeling population growth. Understanding how to solve them is a key skill. So, grab your pencils and let's get started. The core idea behind solving these types of equations is to get both sides to have the same base. Once we achieve that, we can equate the exponents and solve for our variable, which in this case, is 'x'. It's all about manipulating the equation until it's in a form we can easily handle. The process involves using the properties of exponents, some basic algebra, and a bit of patience. Let's start with the basics.

First things first: Understanding the Basics of Exponential Equations. Exponential equations are equations 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. Our goal is to find the value of 'x' that makes the equation true. The key to solving these equations is realizing that if the bases are the same, the exponents must be equal. For example, if we have $2^x = 2^3$, we immediately know that $x = 3$. This is because the exponential function is one-to-one, meaning each input (x) produces a unique output ($2^x$). The critical step, in a situation like the one presented to us, $9^{x-4} = 3^{x+3}$, is to make the bases the same. Looking at our equation, we see that we can express 9 as a power of 3, as $9 = 3^2$. This is our gateway to simplifying the equation. We'll replace the 9 with $3^2$ and then apply the power of a power rule, which is $(a^m)^n = a^{mn}$. The strategy is clear: transform the equation so that both sides have the same base, which makes it simple to solve the equation. So let's convert the bases, use the exponent rules, and simplify them to solve the equation.

Step-by-Step Solution of $9^{x-4} = 3^{x+3}$

Alright, guys, let's get down to business and solve the exponential equation. We'll break it down into several clear steps. Each step will build on the previous one, and by the end, you'll have a clear understanding of how to solve this type of problem. So, here we go!

Step 1: Rewrite with a Common Base

The first step is crucial: rewriting the equation so that both sides have the same base. As we mentioned earlier, we know that $9 = 3^2$. So, we can rewrite the left side of the equation as $(3^2)^{x-4}$. This step is based on the fundamental property of exponents and sets the stage for simplifying the entire equation. Now, our equation looks like this: $(3^2)^{x-4} = 3^{x+3}$. By doing this, we've brought both sides of the equation closer to having a common base, which is what we need to solve for 'x'. The reason we want a common base is that it allows us to equate the exponents. That's the power of the base 3. Always look for a common base when tackling these equations; it will simplify the process significantly.

Step 2: Apply the Power of a Power Rule

Next up, we need to simplify the left side of the equation using the power of a power rule. The power of a power rule states that $(a^m)^n = a^{m*n}$. In our case, we have $(3^2)^{x-4}$. Applying the rule, we multiply the exponents: $2 * (x-4) = 2x - 8$. Now, our equation becomes $3^{2x-8} = 3^{x+3}$. This step is all about tidying up the equation so we can easily compare the exponents. Notice how the equation is becoming much simpler? We're closer to our goal! Just remember to distribute the 2 correctly. Incorrect distribution is a common mistake, so take your time and double-check your work. Now that we have a cleaner equation, we are ready to move on.

Step 3: Equate the Exponents

Now that both sides of the equation have the same base (3), we can equate the exponents. This is where we extract the core of the problem. If $3^{2x-8} = 3^{x+3}$, then it must be true that the exponents are equal: $2x - 8 = x + 3$. This step is a direct consequence of the one-to-one property of exponential functions. Because the bases are identical, the only way for the equation to hold true is if the exponents are equal. This crucial transformation changes the exponential equation into a linear equation, which we can easily solve using basic algebra. So, with this step, we’ve effectively changed the problem into something simple.

Step 4: Solve for x

Alright, now we have a simple linear equation: $2x - 8 = x + 3$. The goal is to isolate 'x' on one side of the equation. First, subtract 'x' from both sides: $2x - x - 8 = x - x + 3$, which simplifies to $x - 8 = 3$. Next, add 8 to both sides to isolate x: $x - 8 + 8 = 3 + 8$, which simplifies to $x = 11$. And there you have it! We've solved for 'x'. We found that the solution to the equation $9^{x-4} = 3^{x+3}$ is $x=11$. Now, we should check our solution to make sure that it's correct.

Step 5: Verify the Solution

It's always a good practice to verify the solution to ensure it's correct. Let's substitute x = 11 back into the original equation, $9^{x-4} = 3^{x+3}$, to check. Substituting x = 11, we get $9^{11-4} = 3^{11+3}$. This simplifies to $9^7 = 3^{14}$. Now, let's rewrite 9 as $3^2$, so we have $(3^2)^7 = 3^{14}$. Using the power of a power rule, this becomes $3^{14} = 3^{14}$. Both sides are equal, which means our solution is correct! Verifying the solution is a vital step because it confirms that our process and calculations are accurate. This step helps us build confidence in our problem-solving skills and ensures that we deliver correct answers.

Conclusion

So there you have it, guys. We successfully solved the exponential equation $9^{x-4} = 3^{x+3}$, step-by-step. Remember, the key is to get the same base on both sides of the equation. Then, use the exponent rules to simplify and solve for 'x'.

In Summary:

• Rewrite the equation with a common base.
• Apply the power of a power rule.
• Equate the exponents.
• Solve for 'x'.
• Verify the solution.

This approach works for a wide variety of exponential equations. Now you're equipped to tackle similar problems with confidence! Keep practicing, and you'll become a pro at solving these types of equations. If you want, you can try solving some other exponential equation problems as well.