Solving Exponential Equations A Step-by-Step Guide For 5^(x+1) - 3 * 5^x = 250

Hey guys! Ever stumbled upon an exponential equation that looks like a monster but is actually a gentle giant? Today, we're going to break down how to solve the equation 5^(x+1) - 3 * 5^x = 250 step by step. Trust me, once you get the hang of it, you'll be solving these like a pro! Let's dive in and make math a little less intimidating and a lot more fun. We will make sure to explore every detail, so you not only get the answer but also truly grasp the methods involved. So, let’s put on our thinking caps and get started!

Understanding the Problem

Before we jump into solving, let’s make sure we understand what we're dealing with. We have an exponential equation, which means our variable x is sitting up there in the exponent. The equation we're tackling is 5^(x+1) - 3 * 5^x = 250. Exponential equations might seem tricky at first, but they follow certain rules that, once mastered, make solving them a breeze. The key here is to recognize that we need to isolate the exponential term or find a way to rewrite the equation into a more manageable form. Think of it as simplifying a complex recipe – we break it down into smaller, digestible steps.

The main challenge in this equation lies in the different exponents attached to the base 5. We have x+1 in one term and x in another. This is where our knowledge of exponent rules comes in handy. Remember, exponential equations are not just about crunching numbers; they're also about understanding the relationships between exponents and bases. So, let’s warm up those algebraic muscles and prepare to transform this equation into something much simpler. Understanding the structure of the equation is the first step towards conquering it, and we're already off to a great start!

Step 1: Simplify Using Exponent Rules

The first trick up our sleeve is using exponent rules to simplify the equation. Remember that a^(m+n) is the same as a^m * a^n. Applying this rule to our equation, we can rewrite 5^(x+1) as 5^x * 5^1. This is a crucial step because it allows us to combine like terms later on. By breaking down the exponent, we're essentially unlocking the equation, making it easier to handle. It's like disassembling a complex machine into its individual parts to see how they fit together.

So, let's rewrite the equation:

5^(x+1) - 3 * 5^x = 250

Becomes:

(5^x * 5^1) - 3 * 5^x = 250

Now, let’s simplify 5^1, which is simply 5. Our equation now looks like this:

(5^x * 5) - 3 * 5^x = 250

Notice how we've managed to express both terms on the left side with 5^x. This is a big win because it sets us up for the next step: combining like terms. Simplifying using exponent rules is a fundamental technique in solving exponential equations, and mastering it will take you a long way in your mathematical journey. So, keep this trick in your toolkit, and let's move on to the next step!

Step 2: Combine Like Terms

Now that we've simplified the exponents, it's time to combine those like terms. Look at our equation: (5^x * 5) - 3 * 5^x = 250. We have two terms that both include 5^x. Think of 5^x as a common factor, just like you would with 'y' in an algebraic expression. We can factor out 5^x, making the equation look much cleaner and easier to work with. Factoring is like organizing your closet – it makes everything more accessible and manageable!

Let's factor out 5^x:

5 * 5^x - 3 * 5^x = 250

This can be rewritten as:

(5 - 3) * 5^x = 250

Now, simplify the expression inside the parentheses:

2 * 5^x = 250

See how much simpler the equation has become? By combining like terms, we've reduced the complexity and brought ourselves closer to isolating the exponential term. This step highlights the power of algebraic manipulation – rearranging and simplifying expressions to reveal the underlying structure. With our equation now in this form, we're ready to isolate 5^x and proceed towards finding the value of x. Let's keep up the momentum and move on to the next step!

Step 3: Isolate the Exponential Term

Our equation currently looks like this: 2 * 5^x = 250. The next logical step is to isolate the exponential term, which in this case is 5^x. To do this, we need to get rid of the coefficient 2 that's multiplying 5^x. How do we do that? Simple – we divide both sides of the equation by 2. This is a fundamental principle of algebra: whatever operation you perform on one side of the equation, you must also perform on the other side to maintain balance. Think of it like a seesaw – if you add weight to one side, you need to add the same weight to the other to keep it level.

So, let's divide both sides by 2:

(2 * 5^x) / 2 = 250 / 2

This simplifies to:

5^x = 125

Voilà! We've successfully isolated the exponential term. Now, the equation is much more manageable, and we're one step closer to solving for x. Isolating the variable or exponential term is a crucial technique in solving equations, and we've executed it perfectly here. This step demonstrates the elegance of algebraic manipulation – using simple operations to transform a complex equation into a straightforward one. Now that we have 5^x isolated, we can move on to the final step: solving for x.

Step 4: Solve for x

We've reached the final stretch! Our equation is now 5^x = 125. To solve for x, we need to express 125 as a power of 5. This is where recognizing patterns and knowing your powers comes in handy. We're looking for a number that, when 5 is raised to that power, equals 125. If you're familiar with powers of 5, you might already know the answer. If not, don't worry – we'll figure it out together. Think of it as solving a puzzle – we need to find the right piece that fits perfectly.

Let's think about the powers of 5:

• 5^1 = 5
• 5^2 = 25
• 5^3 = 125

Aha! We've found it. 125 can be expressed as 5^3. So, we can rewrite our equation as:

5^x = 5^3

Now, we have the same base on both sides of the equation. When this happens, we can simply equate the exponents. This is a fundamental property of exponential equations: if a^m = a^n, then m = n. It's like saying if two trees have the same height, they must have grown for the same amount of time (assuming they grow at the same rate!).

So, we can conclude that:

x = 3

And there you have it! We've successfully solved the equation. The value of x that satisfies the equation 5^(x+1) - 3 * 5^x = 250 is 3. This step showcases the power of recognizing patterns and using the properties of exponents to simplify and solve equations. We've reached the finish line, but the journey of learning mathematics is continuous. Let's do a quick recap of the steps we took to solve this equation.

Conclusion: Recap and Final Thoughts

Wow, guys, we did it! We successfully solved the exponential equation 5^(x+1) - 3 * 5^x = 250. Let’s take a quick look back at the steps we followed:

1. Simplify Using Exponent Rules: We rewrote 5^(x+1) as 5^x * 5^1.
2. Combine Like Terms: We factored out 5^x and simplified the equation.
3. Isolate the Exponential Term: We divided both sides by 2 to get 5^x by itself.
4. Solve for x: We expressed 125 as 5^3 and equated the exponents to find x = 3.

By breaking down the problem into these steps, we transformed a seemingly complex equation into a manageable one. Remember, the key to solving exponential equations is to understand the properties of exponents and use them to simplify the equation. It's like having a toolbox full of handy tools – each rule and technique is a tool that helps you tackle different types of problems.

Solving exponential equations like this is not just about finding the right answer; it’s about building your problem-solving skills and mathematical confidence. Each equation you solve is a step forward in your mathematical journey. So, keep practicing, keep exploring, and remember to have fun with it! Whether you're tackling homework, preparing for an exam, or just curious about math, understanding these techniques will serve you well. Keep up the great work, and who knows what mathematical challenges you'll conquer next!