Solving Exponential Equations 5^x + 125 * 5^(-x) = 30 A Step-by-Step Guide

Hey guys! Today, we're diving into the exciting world of exponential equations. We're going to break down how to solve the equation 5^x + 125 * 5^(-x) = 30. This type of problem might seem intimidating at first, but don't worry! We'll go through it together, step by step, so you can ace these kinds of questions. So, grab your thinking caps, and let's get started!

Understanding Exponential Equations

Before we jump into solving, let's quickly recap what exponential equations are. In simple terms, an exponential equation is an equation where the variable appears in an exponent. For example, in our equation, 5^x, the x is in the exponent. These equations often require a bit of algebraic manipulation and a good understanding of exponent rules to solve. Now, let's dig into the nitty-gritty and equip ourselves with the knowledge needed to tackle this equation head-on. Remember, understanding the basics is key to mastering more complex problems. So, let's get to it!

Key Concepts and Properties

First off, let's talk about some key concepts and properties that will be super helpful. When dealing with exponential equations, remembering a few crucial rules can make all the difference. For starters, the property a^(-n) = 1/a^n is a lifesaver. This means we can rewrite terms with negative exponents as fractions, which often simplifies things. Also, knowing your exponent rules, like a^(m+n) = a^m * a^n and (a^m)n = a^(m*n), is essential for manipulating equations. These rules allow us to combine and separate exponents strategically. Beyond these, recognizing common exponential values (like powers of 2, 3, and 5) can help spot opportunities for simplification. For instance, recognizing that 125 is 5^3 is a game-changer in our equation. Keep these concepts in your mental toolkit, and you'll be well-equipped to handle various exponential challenges. Alright, with our foundation set, let’s move on to the exciting part – solving the equation!

Step-by-Step Solution

Okay, let's get our hands dirty and solve this equation: 5^x + 125 * 5^(-x) = 30. We'll break it down into manageable steps so it's super clear. First, we're going to deal with that pesky negative exponent, and then we'll transform the equation into something a bit more familiar. Trust me, it’s like turning a confusing puzzle into a straightforward one. So, let's roll up our sleeves and dive into the step-by-step process. By the end of this, you'll be solving exponential equations like a pro!

Step 1: Rewrite the Equation

The first thing we're going to do is tackle that term with the negative exponent, 5^(-x). Remember that a^(-n) = 1/a^n, so we can rewrite 5^(-x) as 1/5^x. This makes our equation look a little friendlier:

5^x + 125 * (1/5^x) = 30

Now, to get rid of the fraction, we'll multiply every term in the equation by 5^x. This is a classic algebraic trick to clear out denominators and make things simpler. So, let’s do it:

(5^x) * 5^x + 125 * (1/5^x) * 5^x = 30 * 5^x

This simplifies to:

(5^x)2 + 125 = 30 * 5^x

See? We're already making progress! By rewriting the equation and eliminating the fraction, we've transformed it into a form that's much easier to work with. Next up, we'll make a clever substitution to turn this into a quadratic equation. Stay tuned!

Step 2: Substitution

Now, let's make this equation even simpler by using a substitution. We'll let y = 5^x. This might seem like a small step, but it's a game-changer because it transforms our exponential equation into a quadratic equation, which we know how to solve. So, wherever we see 5^x, we'll replace it with y. This technique is super handy in many math problems, so it’s a great one to have in your toolkit.

Substituting y into our equation, we get:

y^2 + 125 = 30y

Now, let’s rearrange the terms to get a standard quadratic equation form:

y^2 - 30y + 125 = 0

Voilà! We've successfully converted our exponential equation into a quadratic equation. Isn't that neat? Next, we'll solve this quadratic equation to find the values of y. Let’s keep the momentum going!

Step 3: Solve the Quadratic Equation

Alright, we've got ourselves a quadratic equation: y^2 - 30y + 125 = 0. There are a couple of ways we can solve this – factoring, using the quadratic formula, or even completing the square. For this one, factoring looks like the easiest route. We need to find two numbers that multiply to 125 and add up to -30. Can you think of what they might be?

The numbers are -5 and -25, since (-5) * (-25) = 125 and (-5) + (-25) = -30. So, we can factor the quadratic equation as:

(y - 5)(y - 25) = 0

Now, we set each factor equal to zero to find the values of y:

y - 5 = 0 or y - 25 = 0

Solving these, we get:

y = 5 or y = 25

Great job! We've found the values of y. But remember, we're trying to solve for x, so we're not quite done yet. Next, we'll substitute back to find the values of x. Let's keep pushing forward!

Step 4: Substitute Back to Find x

Okay, we've found that y = 5 or y = 25. Now, we need to substitute back to find the values of x. Remember our substitution: y = 5^x. So, we'll plug in our values of y and solve for x. This is where our understanding of exponents will really shine!

First, let's use y = 5:

5^x = 5

Since 5 is the same as 5^1, we have:

5^x = 5^1

This means:

x = 1

Now, let's use y = 25:

5^x = 25

We know that 25 is 5^2, so:

5^x = 5^2

This gives us:

x = 2

Fantastic! We've found two possible solutions for x: x = 1 and x = 2. We're almost at the finish line. Now, let’s summarize our solutions and wrap things up.

Possible Solutions for x

So, after all that awesome work, we've found that the possible solutions for x in the equation 5^x + 125 * 5^(-x) = 30 are:

• x = 1
• x = 2

These are the values that make the equation true. We went from a seemingly complex exponential equation to a clear and simple solution by breaking it down step by step. That's the power of methodical problem-solving! Now, let’s put these solutions in the context of the multiple-choice options you might see in a test.

Matching Solutions to Multiple-Choice Options

In a multiple-choice question, you might see options like:

A) x = 1

B) x = 2

C) x = -1

D) x = 0

Based on our solution, the correct answers are A) x = 1 and B) x = 2. It’s always a great feeling when your hard work pays off and you can confidently choose the right answers! Now that we’ve nailed this problem, let's zoom out and talk about some strategies for tackling similar questions in the future. Knowing how to approach these problems can save you time and stress during exams.

Tips for Solving Exponential Equations

Alright, let's arm ourselves with some killer tips for solving exponential equations. These strategies will not only help you solve equations like this one but also build your confidence in tackling any exponential problem that comes your way. Think of these as your secret weapons in the world of math!

General Strategies

First off, always look for opportunities to simplify. Can you rewrite any terms with a common base? This often involves recognizing powers of numbers (like we did with 125 being 5^3). Also, don't be afraid to use substitution. It's a fantastic way to transform an equation into a more manageable form, like a quadratic equation. Another pro tip is to check your solutions. Plug them back into the original equation to make sure they work. This can save you from making mistakes and boost your confidence in your answer. Finally, practice, practice, practice! The more you solve these types of equations, the better you'll get at recognizing patterns and applying the right techniques. With these strategies in your arsenal, you'll be well-prepared to conquer any exponential equation!

Conclusion

So, there you have it! We've successfully solved the exponential equation 5^x + 125 * 5^(-x) = 30 and found the possible solutions for x. Remember, the key to mastering these types of problems is to break them down into manageable steps, use the right techniques (like substitution and simplification), and practice regularly. With a solid understanding of exponential properties and a bit of algebraic savvy, you'll be solving these equations like a math whiz in no time! Keep up the great work, and happy solving!