Solving Exponential Equations: Find X For 4^x = (1/8)^(x+5)

Hey guys! Let's dive into a fun math problem today that involves solving exponential equations. We're going to figure out the value of x that makes the equation 4^x = (1/8)^(x+5) true. This might seem tricky at first, but don't worry, we'll break it down step by step. Understanding exponential equations is super useful not just for math class, but also for real-world stuff like understanding how investments grow or how populations change over time. So, let’s get started and make sure we nail this concept!

Understanding the Problem

Before we jump into solving, let’s make sure we really understand what the problem is asking. The question presents us with an equation where the variable x is in the exponent. This type of equation is called an exponential equation. Our goal is to isolate x and find the value that satisfies the equation.

To solve 4^x = (1/8)^(x+5), we need to remember a key principle: to solve exponential equations, we often need to express both sides of the equation with the same base. This allows us to equate the exponents and solve for x. Think of it like this: if we have 2^a = 2^b, then we know that a must equal b. This is the foundation of our approach. We will be converting both sides of the equation to have the same base.

Now, let's look closely at our equation: 4^x = (1/8)^(x+5). Can we express both 4 and 1/8 as powers of the same number? Absolutely! Both 4 and 8 are powers of 2. This is our eureka moment! Recognizing this common base is crucial for simplifying and solving the equation. We will rewrite 4 as 2^2 and 1/8 as 2^-3. This transformation will set us up perfectly to solve for x. So, let's move on to the next step where we put this plan into action and start crunching the numbers. Remember, understanding the problem is half the battle, and now we’re well-equipped to tackle the solution!

Rewriting the Equation with a Common Base

Alright, let's get our hands dirty and start rewriting the equation. As we discussed, the key here is to express both sides of the equation using a common base. We identified that both 4 and 1/8 can be written as powers of 2. This is a critical step in solving exponential equations.

First, let’s rewrite 4 as a power of 2. We know that 4 is 2 squared, so we can write 4 as 2^2. This means our left side of the equation, 4^x, becomes (2²⁾x. Using the power of a power rule, which states that (a^m)n = a^(m*n), we simplify (2²⁾x to 2^(2x). So, the left side of our equation is now 2^(2x).

Now, let's tackle the right side of the equation, (1/8)^(x+5). We need to express 1/8 as a power of 2. Remember that 8 is 2 cubed (2^3), so 1/8 is the reciprocal of 2^3, which can be written as 2^(-3). Thus, (1/8)^(x+5) becomes (2^(-3))(x+5). Again, we use the power of a power rule to simplify this. We multiply the exponents: -3 * (x+5) which gives us -3x - 15. So, the right side of our equation is now 2^(-3x - 15).

Putting it all together, our original equation 4^x = (1/8)^(x+5) has been transformed into 2^(2x) = 2^(-3x - 15). See how much cleaner that looks? Now that we have the same base on both sides, we’re in a fantastic position to solve for x. This step of rewriting the equation with a common base is super important, so make sure you're comfortable with it. Next up, we'll equate the exponents and find the value of x!

Equating the Exponents and Solving for x

Okay, guys, this is where the magic happens! Now that we've rewritten our equation with a common base, we can equate the exponents. This is a fundamental principle when solving exponential equations. If a^m = a^n, then m = n. In our case, we've transformed the equation to 2^(2x) = 2^(-3x - 15).

So, we can now confidently say that 2x = -3x - 15. We've turned our exponential equation into a simple linear equation! Now it’s just a matter of using basic algebra to isolate x. Let’s start by getting all the x terms on one side of the equation. We can do this by adding 3x to both sides: 2x + 3x = -3x - 15 + 3x. This simplifies to 5x = -15.

Now, to solve for x, we need to get x by itself. We can do this by dividing both sides of the equation by 5: (5x)/5 = -15/5. This gives us x = -3. Boom! We've found the value of x that satisfies the original equation.

But wait, we're not quite done yet! It’s always a good idea to check our answer to make sure it’s correct. We'll plug x = -3 back into the original equation to verify our solution. This step is super important to catch any mistakes and build confidence in our answer. So, let's move on to the next section where we'll check our solution and wrap things up.

Checking the Solution

Alright, let’s make sure our hard work has paid off! We found that x = -3, but to be absolutely sure, we need to plug this value back into the original equation: 4^x = (1/8)^(x+5). This step is crucial for verifying our solution and making sure we didn’t make any sneaky errors along the way.

Let’s substitute x = -3 into the equation. The left side becomes 4^(-3). Remember that a negative exponent means we take the reciprocal, so 4^(-3) is the same as 1/(4^3), which is 1/64.

Now let's look at the right side of the equation: (1/8)^(x+5). Substituting x = -3, we get (1/8)^(-3+5), which simplifies to (1/8)^2. This means we square 1/8, which gives us (1/8) * (1/8) = 1/64.

Guess what? Both sides of the equation are equal! We have 1/64 = 1/64. This confirms that our solution, x = -3, is correct. Pat yourself on the back – you’ve successfully solved this exponential equation!

Checking our solution not only ensures accuracy but also reinforces our understanding of the problem-solving process. It’s a great habit to get into, especially in math. So, we’ve solved for x, verified our solution, and now we’re ready to wrap things up with a quick recap of what we’ve learned. Let's head to the conclusion!

Conclusion

Fantastic job, everyone! We’ve tackled a tricky exponential equation and come out on top. Let’s recap what we did to solve for x in the equation 4^x = (1/8)^(x+5).

First, we understood the problem and identified that it was an exponential equation. We recognized the key strategy: to express both sides of the equation with the same base. This is the golden rule for solving these types of problems.

Next, we rewrote the equation using the common base of 2. We transformed 4^x into 2^(2x) and (1/8)^(x+5) into 2^(-3x - 15). This step made the equation much simpler to handle. Remember the power of a power rule – it’s your friend here!

Then, we equated the exponents, setting 2x equal to -3x - 15. This turned our exponential equation into a linear equation, which was much easier to solve. We used basic algebraic manipulation to find that x = -3.

Finally, we checked our solution by plugging x = -3 back into the original equation. We verified that both sides were equal, giving us confidence in our answer.

So, the value of x that satisfies the equation 4^x = (1/8)^(x+5) is -3. Awesome work, guys! You've not only solved a challenging problem but also reinforced your understanding of exponential equations. Keep practicing, and you'll become a master of these in no time! Remember, math is all about breaking down complex problems into smaller, manageable steps. You've got this!