Solve It Like A Pro - Exponential Style! 😁

Added: 2026-05-29

[00:00:00]Hello everyone. In this video, we're going to be solving an exponential system.

[00:00:06]We have 5 to the power x equals 4 and 2 to the power y equals 25.

[00:00:14]And we're going to be looking for the values of xy, which is the product. We don't always write the multiplication symbol, especially when two variables are together. It's meant to be multiplication.

[00:00:28]Okay, I'll be presenting two methods, even though I'm pretty sure there is a third or fourth method. Let's go ahead and start with the first method, which is a little bit more logarithmic. Or should I say painful?

[00:00:45]Okay, so we have got 5 to the power x equals 4.

[00:00:48]And I want to log both sides. So you have a couple different options. You can use base 5, you can use base 2 or 4, or you can just use ln. Which one should we use? Let's go with ln because ln is the natural log, it's a special one. And remember Euler's number e to the power x is a special function, which is the base of the natural log, because when you differentiate e to the x, you get the same thing, and that's the only exponential that satisfies it. So it's a very special number. That's not the only special property about it. If you think about the limit as n approaches infinity of 1 + 1 over n to the power n, and then, you know, compound interest and so on and so forth. Anyways, that's another story, probably a few other videos. So, let's go ahead and keep it simple and natural log both sides. ln 5 to the x equals ln 4.

[00:01:42]Awesome. Now, we're going to go ahead and move the x to the front because that's a nice property of logs. x times ln 5 equals ln 4.

[00:01:53]And then from here we get x equals ln 4 over ln 5. Isolating x would be helpful in this case because our goal, always keep that in perspective, we're trying to find xy. So, it would make sense if you could find x and y numerically and then multiply together. Okay?

[00:02:11]So, let's do the same thing for y.

[00:02:13]And from here we got 2 to the y equals 25.

[00:02:17]And the natural log both sides.

[00:02:22]And then bring the y to the front. y * ln 2 = ln 25. And then divide both sides by ln 2.

[00:02:31]And then you'll get y in terms of a quotient like x.

[00:02:36]So, we got these two things. Nice.

[00:02:39]What are we going to do next?

[00:02:41]Multiply them together, right? Because we are looking for xy, right? So, that makes sense.

[00:02:48]x is ln 4 over ln 5.

[00:02:51]Again, ln is natural log. And then y is ln 25 over ln 2.

[00:02:58]Awesome. And then this is a product and you can just multiply, right?

[00:03:03]But is that it? Look, so is this the answer? ln 4 * ln 25 / ln 5 * If you didn't know anything about logarithms, you would probably leave it like that.

[00:03:13]But there is a way to simplify this. So, let's go ahead and simplify this expression. So, here's what we're going to do. First of all, we're going to take advantage of the power rule. And what is the power rule? So, if you have ln x to the n, then you can go ahead and move this and write this as n * ln x. And some people write x in parentheses so that they're not confused, but I'm hoping that you are familiar with this.

[00:03:38]Our function is ln and x is the argument.

[00:03:41]I I don't I don't like writing the parentheses all the time. I'm kind of picky on that. So, ln 4 is understood, I think, but ln x is a little bit uh can more confusing. Like ln x, is that like three letters being multiplied? No.

[00:03:56]Okay, so anyways, so how do you simplify this? We're going to use the power rule.

[00:04:01]Four can be written as two squared and 25 can be written as five squared.

[00:04:07]Great. So we have to basically bring everything down to its prime factors and then we're going to move the powers and then go from there. Move the two to the front, you're going to get two ln two over ln five and move the two to the front two ln five over ln two.

[00:04:26]ln two is going to cancel out, ln five is going to cancel out. Yay, everything cancels out and we end up with four for the value of xy because we were looking for xy and this is it. Makes sense?

[00:04:39]Okay, hopefully it does. So that is the first method. Let's go ahead and take a look at the second method and then we'll finish up. All right? So second method obviously is supposed to be smarter, nicer, shorter.

[00:04:57]So here's how the second method works.

[00:04:58]Let's rewrite the original problem. Five to the x equals four, two to the y equals 25 and what is xy?

[00:05:07]So notice that we got logarithmic expressions for x and y and we multiply them together. Now before we start with the second method I want to I just want to ask you a quick question like pause maybe keep it open-ended.

[00:05:18]If you didn't use ln, what would happen?

[00:05:21]If we use for example log base five and then you know, something like this. Oops, that's supposed to be four, right? Okay, let's go back here. So if we log both sides with base four, then you would get a one here, right? And then it would be like this. Would it matter? Yes, it would be a little harder because you would have different bases, but guess what? You could use change of base formula, but instead of using change of base, we could basically turn everything into ln directly. That way it's going to be a little easier. I find it easier. You can also use log base 10 if you want.

[00:05:56]Anyways, let's continue with the second method. So, we have x um we have these two equations and we're going to do the following.

[00:06:05]We're going to take advantage of the this fact. 4 is 2 squared, so let's go ahead and write it that way. 5 to the x is 2 squared and from the second equation 2 to the y is 5 squared. I know some people are going to say y equals 2 and x equals 2 and multiply it's 4. Even though their product is 4, that is correct, but x does not equal 2 in any way, shape, or form. Now, think about it. Can x be 2 here? x equals 2 means 4 equals 25, which is absolutely nonsense.

[00:06:33]So, never ever say that even though you may arrive at the right answer at the end, you're probably going to get zero points, a big fat zero. Okay, so the method is important. So, instead of that, we're going to do the following.

[00:06:46]We're going to use substitution. But in order to substitute, I need to isolate something. How about isolating the two here?

[00:06:52]The base. So, let's go ahead and raise both sides to the power of 1/2, which is kind of square rooting.

[00:06:58]So, 5 to the x to the power 1/2 is 2 to the 2 to the 1/2. And here the twos are going to cancel out and we're going to end up with the following. 2 equals 5 to the power 1/2 of x.

[00:07:10]Awesome. And then we have 2 to the y equals 5 squared. Now, this two can be replaced with this two. Make sense?

[00:07:21]Like this. So, replace the two with 5 to the power 1/2 of x, raise it to the power y. We're just following the steps here and 5 squared. You see how smooth that is? Obviously, that was the whole purpose for this problem. The first method is kind of like very brute force-y and painful and long, but it's okay. Some people are going to take it and that's perfectly fine.

[00:07:43]The shortest method is not always the best by the way. We're not racing here.

[00:07:47]Trying to learn hopefully. So, now we got the same base. Yay, success. And now we can equate the exponents and that gives us multiplication by two. We get we get xy = 4. And that is the same answer. Yay.

[00:08:04]And this brings us to the end of this video. Thank you for watching. I hope you enjoyed it. Please let me know.

[00:08:08]Don't Please uh I hope you enjoyed the video. Please let me know. Don't forget to comment, like, and subscribe. I'll see you next time with another video. Until then, be safe.

[00:08:19]Take care. And don't forget to watch the shorts. Bye-bye.