Solving 4^x = 1/√2 A Step-by-Step Guide

Aug 3, 2025 by ADMIN 40 views

Solving the Exponential Equation: 4^x = 1/√2

Hey guys! Let's dive into a cool math problem today. We're going to tackle an exponential equation that might look a bit tricky at first, but don't worry, we'll break it down step by step. Our mission is to solve for x in the equation 4^x = 1/√2. Sounds fun, right? Let's get started!

Understanding the Basics

Before we jump into the solution, let's quickly refresh our understanding of exponential equations and some key concepts. An exponential equation is an equation where the variable appears in the exponent. In our case, x is the exponent, and we need to find the value of x that makes the equation true. To solve these types of equations, we often try to express both sides of the equation with the same base. This is because if we have a^m = a^n, then we can confidently say that m = n. This is a fundamental property that will help us immensely in solving our problem.

Another concept we need to keep in mind is how to deal with radicals and fractions in exponents. Remember that the square root of a number can be written as that number raised to the power of 1/2. So, √2 is the same as 2^(1/2). Also, a number in the denominator can be moved to the numerator by changing the sign of its exponent. For example, 1/a^n is the same as a^(-n). These are the little tricks that make solving exponential equations so much more manageable. Keep these in mind, and you'll be solving exponential equations like a pro in no time!

Rewriting the Equation

Okay, so our first step in tackling this exponential equation is to rewrite both sides of the equation using the same base. Looking at our equation, 4^x = 1/√2, we can see that both 4 and 2 are powers of 2. This is a great hint that we should try to express both sides with a base of 2. Let's start with the left side, 4^x. We know that 4 is the same as 2 squared (2^2), so we can rewrite 4^x as (2²⁾x. Using the power of a power rule, which states that (a^m)n = a^(m*n), we can simplify (2²⁾x to 2^(2x). Awesome! We've got the left side in terms of base 2.

Now, let's tackle the right side, 1/√2. As we discussed earlier, we can rewrite √2 as 2^(1/2). So, 1/√2 becomes 1/(2^(1/2)). To get this into the numerator, we can use the rule that 1/a^n = a^(-n). Applying this rule, we rewrite 1/(2^(1/2)) as 2^(-1/2). Fantastic! Now we have both sides of the equation expressed with the same base: 2^(2x) = 2^(-1/2). This is a huge step forward, guys. We've transformed the equation into a form that's much easier to solve.

Solving for x

With both sides of our equation now expressed with the same base, we can finally solve for x. Remember, we have 2^(2x) = 2^(-1/2). The fundamental property we mentioned earlier comes into play here: if a^m = a^n, then m = n. So, in our case, if 2^(2x) = 2^(-1/2), then 2x must equal -1/2. This simplifies our problem to a simple algebraic equation: 2x = -1/2. See how much easier it gets when we break it down?

To isolate x, we need to get rid of the 2 that's multiplying it. We can do this by dividing both sides of the equation by 2. So, we have (2x)/2 = (-1/2)/2. On the left side, the 2s cancel out, leaving us with x. On the right side, dividing -1/2 by 2 is the same as multiplying -1/2 by 1/2, which gives us -1/4. Therefore, our solution is x = -1/4. We've found the value of x that satisfies the original equation! This is what it's all about, guys. We took a seemingly complex exponential equation and, using some basic rules and properties, solved for the unknown. Go us!

Verifying the Solution

It's always a good practice to verify our solution to make sure we didn't make any mistakes along the way. To verify our solution, we'll plug x = -1/4 back into the original equation, 4^x = 1/√2, and see if it holds true. Substituting x = -1/4, we get 4^(-1/4) = 1/√2. Now, let's simplify the left side.

We can rewrite 4 as 2^2, so 4^(-1/4) becomes (2²⁾(-1/4). Using the power of a power rule, we get 2^(2*(-1/4)), which simplifies to 2^(-1/2). Remember that a negative exponent means we take the reciprocal, so 2^(-1/2) is the same as 1/(2^(1/2)). And, as we know, 2^(1/2) is the same as √2. So, 1/(2^(1/2)) is equal to 1/√2. Awesome! The left side of the equation simplifies to 1/√2, which is exactly what the right side is. This confirms that our solution, x = -1/4, is correct. High five!

Alternative Methods

While we've solved the equation quite nicely by expressing both sides with the same base, it's worth mentioning that there are other methods we could have used. One such method involves using logarithms. Logarithms are essentially the inverse operation of exponentiation, and they can be very powerful tools for solving exponential equations. If we had taken the logarithm of both sides of the original equation, we could have used logarithmic properties to isolate x. This method can be particularly useful when it's not straightforward to express both sides with the same base.

Another approach could involve directly dealing with the fractional exponent and the radical. We could have tried to rationalize the denominator on the right side or manipulate the exponents using different exponent rules. However, the method we used, expressing both sides with the same base, is often the most straightforward and intuitive approach for this type of problem. But hey, it's always good to know there are multiple paths to the solution, right? It keeps things interesting and helps us develop a deeper understanding of the math involved.

Practice Problems

To really solidify your understanding of solving exponential equations, it's essential to practice, practice, practice! Here are a few problems similar to the one we just solved. Give them a shot, and you'll be an exponential equation master in no time!

9^x = 1/√3
25^x = 1/5
8^x = 1/√4

Try to solve these using the method we discussed, expressing both sides with the same base. Remember to rewrite radicals as fractional exponents and use the power of a power rule. And, of course, always verify your solutions. You got this!

Conclusion

So, guys, we've successfully solved the exponential equation 4^x = 1/√2 and found that x = -1/4. We walked through the process step by step, from rewriting the equation with the same base to solving for x and verifying our solution. We also touched on alternative methods and provided some practice problems to help you hone your skills. Solving exponential equations might seem daunting at first, but with a solid understanding of the basic rules and properties, you can tackle them with confidence. Keep practicing, keep exploring, and keep enjoying the world of math! You're doing great, and I'm excited to see what other math challenges you conquer. Until next time, keep those equations balanced and those exponents in check!