I Posted This Exponential Equation on Facebook and Most People Failed

Hello everyone. In today's video, we're going to work out a problem that I put on my Facebook page. This is an equation. And one person responded on Facebook to this. I got a bunch of likes, but only one person answered it. Raj Shrestha answered it on Facebook and he got it right. He got it completely right. And so we're going to work it out in this video. I'm going to show you how to solve this equation. And if you like problems like this, you know, maybe I'll post more here. But this was a primarily a Facebook question. And so I thought, well, I've already got the problem. Might as well provide the solution and make a video. So that's why I'm doing it because I felt like well, I should solve it. You know, if I'm going to post it on Facebook, I should work it out. So let's when you see something like this, your natural inclination might be to write it like this. And then the next thing you might want to do is clear the fractions. So if you do that, you end up with 2 to the x times 2 to the x, which is 2 to the x quantity squared, right? There's a one here. 2 to the x times 1 over 2 to the x is 1 and that's 5 times 2 to the x. This is not my first instinct. And the reason is um I've done problems like this before, so right away I was just thinking multiply by 2 to the x right away. And when you do that, without thinking of it this way, it still works. To show you two ways to approach it. You get 2 to the x squared. Right? You add the exponents. So I went straight here. That was my first inclination. So I skipped all of this. I just did it in my head. But you can show as much work as you want. You can see multiple ways to show work here to arrive at conclusions. So there's multiple ways to do it. This is a quadratic equation in 2 to the x. So we're going to set everything equal to zero. We'll subtract 5 times 2 to the x from both sides. So we have 2 to the x quantity squared minus 5 times 2 to the x plus 1 equals 0. Okay. And so we'll make a substitution. We're going to put or set u equal 2 to the x. So we end up with a quadratic equation in u. Makes it easier to think about. u squared minus 5 u plus 1 equals 0. And the majority of people, you know, the majority of students would use the quadratic formula cuz that's easier. Um I don't want to use that. I'm going to complete the square cuz that's just faster for me. So let's go ahead and complete the square. So when you're completing the square, the first thing you want to do is isolate all the variable terms. So we're going to subtract one from both sides. So you have u squared minus 5 u equals -1. And so when you're completing the square, what you do is you take the coefficient of u in this case, which is -5. You divide it by two and you square it. So this is -5 over 2 quantity squared, which is 25 over 4. So now we simply add this to both sides, so we get u squared minus 5 u plus 25 over 4 equals -1 plus 25 over 4. And you can add these pretty easily. This is -4 over 4 plus 25 over 4, which is 21 over 4. So writing it again. Going kind of fast here. My nose is a little clogged. I was riding a motorcycle and I didn't have the face mask on and I think it really hurt my sinuses, but hopefully I'm getting better. I'm not sick, but my nose is a little clogged today. So we've completed the square. What does that mean? This will factor. This is called the perfect square trinomial, so it will always factor. You just take the coefficient of u and divide it by two. Take the square root of both sides. So we get u minus 5 halves equals plus or minus the square root of 21 over 2. Right? Cuz the square root of 4 is 2. Add 5 halves to both sides. So you have u equals 5 halves plus or minus the square root of 21 over 2. But we are not done, my friends. Right? We are not done because our original variable our original variable was x. Right? So or rather 2 to the x. u was 2 to the x. So this is 2 to the x. Well, the variable is x, but u is 2 to the x. And then we get here, we can convert this from exponential form to logarithmic form. You can't take the natural log of both sides, but I'm not going to do that. I'm just going to do a straight conversion. The logarithm is the exponent. The base is two. And this requires some caution. That's the answer. You can write it as a single fraction and you can break it up using properties of logarithms as well. In fact, the person on Facebook who did it broke it up using properties of logs. Let me show you. This is 5 plus or minus the square root of 21 over 2. And then you can break this up using the quotient rule for logarithms, but I'll just leave it. There is a word of caution here in problems like this. So you look at the square root of 21. The first thing you should be thinking about is does this make sense? All right? Because if you think about the graph of log base 2 of x, well, the domain of this graph includes only positive numbers. In fact, it's the set of positive numbers. That is the domain. Uh it zero is not included and negative numbers are not in the domain of this function. We're dealing with a real valued logarithmic function. And so you got to make sure that this is positive. And so the way to do that is to look at the square root of 21 and note that it's less than the square root of 25, which is 5. Right? And so you know that then that 5 halves go over here. 5 halves plus or minus the square root of 21 minus the plus doesn't matter. The plus is positive. But the minus is what matters here. The term with the minus. This is going to be greater than zero because this number here is less than 5. Right? So we're okay. You got to be really careful, right? Because you cannot have all kinds of problems in this video. This is my sharpener is failing. You cannot have the log of a negative number. All right? And that's no good. So you got to be really careful, my friends, uh when you're doing uh problems like this. And let me check Facebook. My Mac turned off here. Go. Looks like yeah, one person got it right. It got seven likes and a couple shares. And one person answered it and got it right. And I think the reason a lot of people didn't answer this question is because they couldn't do it in their heads. Right? I mean, who's going to come up with this answer in their head? Uh so it's a little bit harder than your typical Facebook math problem, but I thought I should work it out cuz you know, I posted it on Facebook. And I had the problem here, so I'm like, well, I've already got the problem and I've already got the picture from Facebook. I'll just make that the thumbnail and throw it on YouTube. Why not, right? But kind of an interesting problem. This is something that you might see like a precalculus course. When I've taught precalculus in the past, this is something that I might put on a test. You know, it'd be like a regular test Uh but yeah, hope it's been helpful. And hey, if you want to learn math, check out my books and courses. All those links are in the description. Stay strong, my friends.