Finding 4^n: A Step-by-Step Solution To Exponential Equations

Hey everyone! Let's dive into a fascinating math problem today: finding the value of 4^n, where 'n' is the product of the roots of the equation 16^x - 3 * 4^x = 4^(x+5). This problem looks complex, but we'll break it down step by step to make it super easy to understand. We'll explore the fundamental concepts of exponential equations, how to manipulate them, and finally, how to arrive at the solution. So, grab your calculators and let's get started!

Understanding the Exponential Equation

At the heart of this problem lies an exponential equation. Exponential equations are equations where the variable appears in the exponent. Our equation, 16^x - 3 * 4^x = 4^(x+5), is a classic example. To solve it, we need to use some clever algebraic manipulation and a bit of insight into the properties of exponents.

Before we jump into the solution, let’s recap some key concepts. Remember that exponents indicate how many times a base number is multiplied by itself. For instance, 4^2 means 4 multiplied by itself (4 * 4), which equals 16. Similarly, 16^x means 16 multiplied by itself 'x' times. The challenge here is that 'x' is unknown, and we need to find its value (or values) to solve for 'n' and eventually find 4^n.

The key to solving exponential equations is often to express all terms with the same base. Why? Because if we have the same base, we can equate the exponents. This simplifies the equation considerably. In our case, we can express both 16 and 4 as powers of 4, which will be our first step in simplifying the equation. Keep this strategy in mind, guys, as it's a lifesaver for many exponential equation problems!

Step 1: Simplifying the Equation

The first step is to simplify the given equation: 16^x - 3 * 4^x = 4^(x+5). To do this, we'll express all terms as powers of 4. Remember, 16 is the same as 4^2. So, we can rewrite 16^x as (4²⁾x. Using the rule of exponents that says (a^b)c = a^(b*c), we can further simplify this to 4^(2x). This is a crucial move because it gets us closer to having a common base for all terms in the equation.

Now, let’s look at the right side of the equation: 4^(x+5). Using another rule of exponents, a^(b+c) = a^b * a^c, we can rewrite this as 4^x * 4^5. This step is essential because it separates the exponent into manageable parts, making it easier to work with.

So, after these simplifications, our equation now looks like this: 4^(2x) - 3 * 4^x = 4^x * 4^5. We're making progress! By expressing all terms as powers of 4 and breaking down the exponents, we’ve transformed a seemingly complex equation into a more approachable form. Next, we’ll use a substitution to make the equation even simpler to handle. Hang in there, guys, we're on the right track!

Step 2: Substitution for Simplicity

To make our equation even easier to handle, we'll use a clever trick called substitution. Let's substitute y = 4^x. This might seem like a small step, but it's going to make a big difference in how we perceive the equation. By replacing 4^x with 'y', we're essentially transforming our exponential equation into a quadratic equation, which we know how to solve.

So, if y = 4^x, then 4^(2x) is the same as (4^x)2, which is y^2. Now, let's substitute 'y' into our simplified equation from the previous step: 4^(2x) - 3 * 4^x = 4^x * 4^5. Replacing 4^(2x) with y^2, 4^x with y, and calculating 4^5 (which is 1024), our equation becomes: y^2 - 3y = 1024y. See how much simpler it looks?

This substitution is a powerful technique in solving exponential equations. It allows us to convert a complex-looking equation into a more familiar form that we can easily solve using standard algebraic methods. Now that we have a quadratic equation, we can rearrange it into the standard form and find the values of 'y'. Let’s move on to the next step where we'll solve for 'y' and then backtrack to find 'x'. You're doing great, guys!

Step 3: Solving the Quadratic Equation

Now that we have our equation in terms of 'y': y^2 - 3y = 1024y, it's time to solve for 'y'. The first thing we need to do is rearrange the equation into the standard quadratic form, which is ay^2 + by + c = 0. To do this, we'll subtract 1024y from both sides of the equation. This gives us: y^2 - 3y - 1024y = 0, which simplifies to y^2 - 1027y = 0.

Now we have a quadratic equation in standard form, where a = 1, b = -1027, and c = 0. To solve this, we can factor out a 'y' from the left side of the equation: y(y - 1027) = 0. This is a fantastic simplification because it immediately gives us two possible solutions for 'y'.

According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. So, either y = 0 or y - 1027 = 0. This means our solutions for 'y' are y = 0 and y = 1027. These are the values of 'y' that satisfy our quadratic equation. But remember, we're not trying to find 'y'; we're trying to find 'x'. So, we need to backtrack and use our substitution (y = 4^x) to find the corresponding values of 'x'. Let's tackle that in the next step, guys!

Step 4: Finding the Values of x

We've found the values of 'y', which are y = 0 and y = 1027. Now, we need to use our substitution, y = 4^x, to find the corresponding values of 'x'. This step is crucial because 'x' is directly related to 'n', which we need to calculate 4^n.

Let's start with the first value, y = 0. We have 4^x = 0. Now, think about this: is there any value of 'x' that will make 4 raised to that power equal to zero? The answer is no. Any positive number raised to any power will never be zero. So, y = 0 does not give us a valid solution for 'x'. This is an important observation because it eliminates one potential root, simplifying our problem.

Now, let's consider the second value, y = 1027. We have 4^x = 1027. To solve for 'x', we need to use logarithms. Taking the logarithm base 4 of both sides, we get: x = log_4(1027). This is the value of 'x' that corresponds to y = 1027. We can use a calculator to find the approximate value of x, but for the purpose of finding 'n', we'll keep it in this form for now.

So, we have one valid solution for 'x': x = log_4(1027). Since our equation is derived from a simplified form, it is important to check for extraneous solutions. In this case, substituting x = log_4(1027) back into the original equation will confirm that it is indeed a valid solution. Now that we have the value of 'x', we can move on to finding 'n', which is the product of the roots. Let’s do that next, guys!

Step 5: Calculating 'n'

We've determined that we have one valid solution for x: x = log_4(1027). The problem states that 'n' is the product of the roots of the equation. Since we only have one valid root, the product of the roots is simply the value of that root itself. Therefore, n = log_4(1027).

Now that we have 'n', our final step is to find the value of 4^n. This is where everything comes together! We have 4^n, and we know that n = log_4(1027). So, we need to calculate 4^(log_4(1027)).

Remember the logarithmic property that states a^(log_a(b)) = b. This is a crucial property that will help us simplify our expression. In our case, 'a' is 4, and 'b' is 1027. So, 4^(log_4(1027)) is simply equal to 1027. This is an elegant simplification that directly gives us the answer we've been looking for.

Therefore, 4^n = 1027. We've successfully found the value of 4^n by simplifying the equation, using substitution, solving the quadratic equation, finding the roots, and applying logarithmic properties. You've done an amazing job following along, guys! Let's wrap up with a final conclusion.

Conclusion

In conclusion, we've successfully navigated the problem of finding the value of 4^n, where 'n' is the product of the roots of the equation 16^x - 3 * 4^x = 4^(x+5). By breaking down the problem into manageable steps, we simplified the equation, used substitution to transform it into a quadratic equation, solved for the roots, and applied logarithmic properties to find that 4^n = 1027.

This problem highlights the importance of understanding exponential equations, algebraic manipulation, and logarithmic properties. These are fundamental concepts in mathematics that are essential for solving a wide range of problems. You've gained valuable skills by working through this problem, guys, and you can apply these techniques to other mathematical challenges.

So, keep practicing, keep exploring, and keep having fun with math! You've got this!