3ˣ = 20 | Can You Solve This Exponential Equation?

[00:00:00]Okay, so let's go ahead and get going here.

[00:00:02]So, 3 to the x power is equal to 20.

[00:00:05]Okay, now what type of equation is this?

[00:00:08]So, in algebra, you need to identify what type of equation you're dealing with because there's all sorts of different type of equations, right?

[00:00:16]Polynomial equations, rational equations, radical equations, systems of equations, linear equations. I can go on and on, quadratic equations. So, this particular equation, what kind of gives us a hint here is where the variable is at. Okay, the unknown value is in the exponent. So, this type of equation is what we call an exponential equation.

[00:00:39]Okay, now what is the objective here?

[00:00:41]Well, we're trying to find the solution, which is the value of x that makes this equation true. Okay?

[00:00:49]So, we can kind of think about this in this manner, right? So, here is our problem. 3 to the x power is equal to 20. So, let's just kind of run some basic experiments here. You know, say what All right, what value of x, 3 to the what power is going to be 20? Well, let's just start running some numbers, right? So, 3 to the first power, well, that's 3. Well, that's certainly not 20, so we're going to have to increase our power here, right? So, we'll bump it up to 2.

[00:01:16]Now, 3 to the second power is 3 * 3, that's 9. Okay, well, that's better than 3. It's getting closer to 20, but we need to keep going, right? So, how about 3 to the third power? Well, that's 3 * 3 * 3, that's 27. Well, now we went too much, right? So, 3 to the third power is 27. We're looking for 3 to the some power, kind of squeeze it in right here, is equal to 20. Okay, so what can we kind of conclude here? Well, hopefully you can see that our power is going to be greater than 2, but less than 3. And probably a little bit closer to 3. So, if I was to to take a guess, maybe like 2.75.

[00:01:58]Now, what someone could do just by using common sense is just to kind of do a guess and check method and just, you know, keep testing numbers until they get pretty close to 20, right? 3 to some power. Of course, that's going to be some sort of decimal till you get pretty close to 20, but that is kind of a common sense guess and check approach.

[00:02:18]But, what we don't want to do is just be like, "Well, I'll just kind of guess and check in order to solve a problem like this." Well, no, we don't want to do that. We want to be precise in our answer and we're going to get into that right now.

[00:02:30]Okay, so again, we're dealing with an exponential equation. Now, an exponential equation, remember, is where you're trying to solve for the variable and the variable is in the exponent location of a power. Now, what you need to know is that when you have an exponential equation, you're going to be using logarithms. Okay? Now, on your calculator, if you have a scientific calculator, uh and you're just interested in this, you want to look for this button, the LOG button. Now, there's another button called LN. This is what we call the natural logarithm.

[00:03:02]This is the common logarithm. And typically, for most problems, you'll use uh the LOG button. You could use both buttons. There is a a specific uh kind of situation where you use a natural log. Again, I don't want to go on uh too many tangents here because this is a big discussion uh in algebra. Now, if you're saying, "Yes, yes, Mr. YouTube Math Man, this is what I'm studying in my math class and I need help." Uh for example, if you're taking like college algebra, well, I'm going to go ahead and leave uh links to my uh algebra courses in the description of this video. You want to check out algebra 2 or precalculus, okay? So, I teach this in both. It all depends on where you're at, but if you want to, you know, formal instruction from me, check that out. Um also, I have a ton of additional videos on this stuff on my YouTube channel as well. Okay. So, when you have exponential equations, you're going to use logarithms to solve those. And when we have a logarithmic equation, you're going to use uh exponents. And of course, you you might not even know what a logarithmic equation is. It's probably something that has, you know, like uh a log in it, right? In some sort of equation, something like this. You're going to use x uh exponents or powers. That's because exponential functions and logarithmic functions are inverses of one another.

[00:04:19]All right? Okay, so again, if you never learned this before, I'm giving you really good, you know, quick summary of what this topic is about and you'll be have pretty good um you know, foundation to build upon uh you know, after watching this video, right? So, I'm going to kind of keep it simple. So, all right. So, we have this exponential equation. So, we're going to want to use logarithms to solve. All right, so how do we do that? Well, let's go ahead and first make sure you understand what a logarithm is.

[00:04:49]Okay. Now, let me draw your attention right over here, okay? So, this thing right here, I would say this in mathematics, this is log base 2 16 is equal to 4. Now, some of you are saying, "Well, what does that all mean, Mr. YouTube Math Man?" Well, I'm going to give you a lovely little saying right here, okay? Now, you ready? Here is the saying. It's bacon and eggs. Bacon and eggs. And what is this guy? My this guy is off his rocker. Oh, boy, I'm going to have to unsubscribe from his YouTube channel. So, but anyways, listen. If you can remember logarithm bacon and eggs, then you got this. Now, what does that mean? BAE? Well, log b a = e. Bacon and eggs. So, uh what I'm talking about here is the following. I'll get back to this in a second. The b is a base, okay? So, let's just look at this right here.

[00:05:42]2 to the fourth power, okay? Where is the exponent and where is the base?

[00:05:47]Well, the big two down here is the base.

[00:05:49]The little four up here is the exponent and the answer when I take two to the fourth power, the answer is 16. Okay, so a base to an exponent is equal to the answer. Right? So, when you have bacon and eggs, it's log base. The base is a B. This is the answer. That's equal to the exponent.

[00:06:10]So, in this particular example right here, two to the fourth power, that's equal to 16 cuz two times two times two times two, two times itself four times is 16.

[00:06:20]So, let's express this power, okay, using exponents, all right, as a logarithm. All right, so we're just going to follow the bacon and egg approach here. So, this is going to be log. What's the base? Now, we have to, you know, focus in here. The base is two, so I'm going to write that right there, okay. Uh A, what's the answer?

[00:06:42]That's 16. So, I'll write that right there and that's going to be equal to the exponent, which of course is four.

[00:06:50]Now, you know, at first, this is going to be confusing because it's new to you, but if you just take your time, you'll be able to write logarithmic expressions as power expressions, exponential, you know, expressions using exponents, power expressions, okay? And you're going to have to be able to go from one to another, okay? So, again, as long as you can remember bacon and eggs and the base to an exponent is equal to the answer, then you'll be good to go. Okay, so now that you understand logarithms and you understand that we're going to need to use logarithms to solve an exponential equation, then we can actually solve this problem.

[00:07:27]Okay, so here is the problem, okay?

[00:07:30]Here's where we kind of first, you know, we're looking at the problem like three to the X. What is this equal to? We kind of did a little experiments, three to the first, that's not going to work.

[00:07:39]Three to the second, Uh is 9 3 cubed, that's 27. So, what is how can we solve for X? Well, now with our new knowledge, uh we can say, "All right, this is an exponential equation, so I've got I have to use logarithms."

[00:07:56]And now I'm going to teach you how you use logarithms to solve an exponential equation. Okay, so the first step is and I'm kind of skipping over a lot of things here because these um problems can be much more complicated.

[00:08:10]But effectively, when you have a power and and a base and exponent on one side and a number on another side, just like this uh very simple situation, at this point, what we can do is take the log of both sides. The log of both sides. That's l o g, that's a logarithm of both sides.

[00:08:29]Now, here is what I want you to understand. If you see something like log of 17 or log of 20. Now, this looks complicated, right? Like, "Oh my god, this this is advanced math." No, no, this is just a number. It's a decimal, okay? And if you do have a calculator handy, a scientific calculator, you can just go on your uh phone and you can probably uh use one of your your calculator app and put it into scientific mode, but you'll just see you can put in uh numbers, take the l o g of it. This is just a decimal, okay?

[00:08:59]So, don't panic and stuff. All we're doing here is take the taking a logarithm of these numbers, okay? All right, so we're going to take the log of both sides. Now, do not do anything yet with your calculator, okay? Do not do anything. We'll do that at the very end.

[00:09:15]But uh what we're going to do is take the log of both sides, okay? Remember, the objective here is to solve for X.

[00:09:21]Now, when you study logarithms, okay, you uh there's some properties of logarithms that you need to know. And one of the best ones, uh they're all important, of course, is you can take when you have the when you have the log of both sides, when you take the logarithm of a of a power, okay, like this, this exponent up here, you can actually drop this exponent right here in front of the log. Okay, so when I took log of 3 x, I can actually take this exponent x and put it right in front of the log. So now this is x log 3 is equal to log 20. Now, at this point, this is a basic algebra problem, right?

[00:09:59]Uh some of you might say, "Well, it might be basic for you, Mr. YouTube math man, but this is not basic for me."

[00:10:04]Well, listen. Remember I told you that the log of a number is just a decimal, okay? This is just a decimal you can get in your calculator. This is another decimal you can get in your calculator, and we're solving for the variable x.

[00:10:16]So if you think about it, this is just x times some number, let's say like seven, is equal to another number like say 14.

[00:10:23]So if I had x 7 is equal to 14, we don't write things like this in algebra, right? We put the variable behind. So this is nothing more than 7 x is equal to 14. So to solve for x, I just got to divide both sides of the equation by 7, okay? And hopefully you're going to say, "Oh, okay, I understand." And now let's change out to a happy face. I know what I have to do. And do I have to divide both sides of the equation by log 3 to solve for x?" Yes, indeed. Okay, that's what we have to do. And let's go ahead and see the uh the rest of this problem.

[00:10:56]Okay, so now, knowing that this is a decimal, this is a decimal, to solve for x, all I have to do is divide both sides of the equation by log 3, and you're going to end up with this expression, log 20 over log 3.

[00:11:09]Now, uh for many of you out there, uh depending on what kind of, you know, class you're in, or if your teacher doesn't want you to use a calculator, this would be the exact correct answer, okay? Uh most teachers would be happy with this, but we want to actually calculate this.

[00:11:24]So in your calculator, you're obviously going to use the log button. That's log base 10, okay? That's what we call the common log. You could use the ln button.

[00:11:34]That is log base e. Okay, base e e is a number in mathematics one of the most important numbers in math. But again, a lot of little tangents here.

[00:11:46]You know, you know, I don't want to get into the stuff you're going to really, you know, need some full instruction on this. But what you need to understand is that we do need our log button. Okay?

[00:11:57]So, if you go into your calculator, just go log log 20 divided by log of 3, you'll get approximately 2.72. There's some other digits here, but we'll just kind of round us off. So, that's an approximation. Now, some you folks out there, you know, I would say you probably have to be well over 50. 50, 60, for sure.

[00:12:18]Yeah, probably 60 and beyond. Maybe in your late 50s. What am I getting at?

[00:12:23]Well, back in the good old days, just so you some of you younger folks can appreciate, when you studied a book like, let's say, algebra 2, okay? Let's say this is a old school 1960s version, 19 early 70s version of algebra 2. Guess what? You didn't have a calculator back in those days.

[00:12:44]Like, when the first basic calculators came out, I think I got early 70s, they were like, you know, $10,000 and some crazy price like that.

[00:12:52]$5,000. Only engineers and stuff really had those. So, those folks that used to have to do this level of math, you would go into the back of your textbook and they had a bunch of tables. Okay? You just go in the back of it and they they worked perfectly fine and you could kind of just look up log 20 to get a decimal of that and you put right there, you know, you just get that decimal approximation.

[00:13:16]Log 3, same type deal. So, you you know, you would have to reference all these tables. You wouldn't need your book.

[00:13:23]Okay? Now, that also hold true with all kinds of trigonometric functions. Everything that your calculator does for you now used to have to reference your book back in the good old days. Now, there is this other thing called the slide rule, and the slide rule is amazing, okay?

[00:13:40]I have some basics understanding and skill of it. You really For those folks who know how to use a slide rule, that's super impressive. That's like a class in and of itself, but some of the more advanced slide rules, okay? And again, this is stuff 50s, 40s, 60s, or even probably up to the early 70s.

[00:14:00]These are things that are They're basically like a ruler with a little thing, and there's all kinds of numbers here on the side, and you would move this back and forth, and you could calculate something like this using your slide rule, okay? Engineers and whatnot had very kind of elaborate, detailed slide rule.

[00:14:17]So, those were amazing. So, anyways, I bring this up as a little bit of math history. For some of you out there that, you know, you just don't understand how much more math was involved to learn back in the good old days cuz to do your calculations, you just didn't have the luxury of a calculator. Now, calculators are awesome. I definitely love calculators. I have a whole bunch of them, and you need to know how to use your calculator, but you know, sometimes I think you know, with too much technology, I think it does distract from the principles and concepts that you need to understand.

[00:14:53]Okay, so hopefully, you know, you learned a little bit of something here about exponential equations. And again, if you haven't studied this yet, you will study this later on when you get to, you know, like second year algebra.

[00:15:07]Okay, so with all that being said, I definitely wish you all the best in your mathematics adventures. Thank you for your time, and have a great day.