Solving For X: 3^x + 25 * 3^x = 36 | Math Problem

Hey guys! Let's dive into a fun math problem today where we need to determine the natural number x that satisfies the equation 3^x + 25 * 3^x = 36. This might seem a bit daunting at first, but don't worry, we'll break it down step by step and make it super easy to understand. Our main goal is to isolate x, and to do that, we'll use some basic algebraic principles and properties of exponents. So, grab your thinking caps, and let's get started!

Understanding the Equation

Before we jump into solving, let's take a closer look at our equation: 3^x + 25 * 3^x = 36. We can see that we have two terms on the left side, both involving 3^x. This is a crucial observation because it allows us to simplify the equation. Think of 3^x as a single variable for now – it’s something we can factor out. Recognizing these patterns is super important in math because it helps us make complex problems simpler.

Combining Like Terms

So, how do we simplify this? Well, we can combine the like terms on the left side of the equation. We have 3^x and 25 * 3^x. If we factor out 3^x, it's like saying we have one 3^x plus twenty-five 3^x. This is basic algebra, and it’s super useful here. When we combine them, we're essentially adding the coefficients (the numbers in front of 3^x). So, we have:

1 * 3^x + 25 * 3^x = (1 + 25) * 3^x

Which simplifies to:

26 * 3^x

Now our equation looks much cleaner:

26 * 3^x = 36

This is a significant step forward. We've reduced the equation to a simpler form where we can more easily isolate 3^x. Remember, math is all about breaking down problems into manageable pieces, and this is a perfect example of that!

Isolating 3^x

Now that we have 26 * 3^x = 36, our next step is to isolate 3^x. To do this, we need to get rid of the 26 that’s multiplying 3^x. How do we do that? Simple! We divide both sides of the equation by 26. This is a fundamental algebraic operation – whatever we do to one side of the equation, we must do to the other to keep it balanced. So, let’s do it:

(26 * 3^x) / 26 = 36 / 26

On the left side, the 26s cancel out, leaving us with:

3^x = 36 / 26

Now, let's simplify the fraction on the right side. Both 36 and 26 are divisible by 2, so we can reduce the fraction:

36 / 26 = 18 / 13

So, our equation now looks like this:

3^x = 18 / 13

We're getting closer! We've successfully isolated 3^x. But now, we need to figure out what x actually is. This is where we start thinking about how exponents work and how we can solve for a variable in the exponent.

Solving for x Using Logarithms

Alright, we've reached a point where we have 3^x = 18 / 13. Now, how do we get that x out of the exponent? This is where logarithms come to the rescue! Logarithms are the inverse operation of exponentiation. Think of them as the tool we use to “undo” an exponent. There are different types of logarithms, but the most common ones are the natural logarithm (ln) and the base-10 logarithm (log). For this problem, we can use either, but let's go with the natural logarithm (ln) because it's super versatile.

Applying the Natural Logarithm

To solve for x, we'll take the natural logarithm of both sides of the equation. Remember, what we do to one side, we do to the other:

ln(3^x) = ln(18 / 13)

Now, here's where the magic of logarithms really shines. There's a property of logarithms that says ln(a^b) = b * ln(a). In simple terms, we can bring the exponent down and multiply it by the logarithm of the base. Applying this property to our equation, we get:

x * ln(3) = ln(18 / 13)

See how the x is no longer in the exponent? We're making progress!

Isolating x

Now, to isolate x, we need to get rid of the ln(3) that's multiplying it. Just like before, we'll do this by dividing both sides of the equation by ln(3):

(x * ln(3)) / ln(3) = ln(18 / 13) / ln(3)

On the left side, the ln(3) terms cancel out, leaving us with:

x = ln(18 / 13) / ln(3)

We've done it! We've solved for x. Now, all that's left is to calculate the value using a calculator.

Calculating the Value of x

We've found that x = ln(18 / 13) / ln(3). Now, let’s plug this into a calculator to get a numerical value. Make sure your calculator is in radian mode for natural logarithms.

First, calculate ln(18 / 13). This is the natural logarithm of the fraction 18/13. Using a calculator, you should get approximately:

ln(18 / 13) ≈ 0.3285

Next, calculate ln(3). This is the natural logarithm of 3. Using a calculator, you should get approximately:

ln(3) ≈ 1.0986

Now, divide the first result by the second:

x ≈ 0.3285 / 1.0986

x ≈ 0.2990

So, we've found that x is approximately 0.2990. But hold on a second!

Checking for Natural Number Solutions

The original question asked for a natural number solution for x. A natural number is a positive whole number (1, 2, 3, and so on). Our calculated value for x is approximately 0.2990, which is not a natural number. This means there is no natural number solution to the equation 3^x + 25 * 3^x = 36.

Why is this important?

It’s super important to always check your solutions against the original problem’s conditions. In this case, we were looking for a natural number, and our answer didn't fit the bill. This could happen for a few reasons: maybe there’s no solution, or maybe we made a mistake along the way. Always double-check your work and make sure your answer makes sense in the context of the problem.

Conclusion

So, guys, we tackled a cool math problem today! We started with the equation 3^x + 25 * 3^x = 36 and went through the steps of simplifying, isolating terms, and using logarithms to solve for x. We found that x ≈ 0.2990, but since we were looking for a natural number solution, we concluded that there is no natural number x that satisfies the equation. Remember, math is a journey, and it’s all about breaking down problems, understanding the steps, and always checking your answers. Keep practicing, and you'll become a math whiz in no time!