Solving 5^x+5^x=25, don't get this wrong!

Hinzugefügt: 2026-05-15

[00:00:00]Just a nice equation. We'll be solving 5 to the x + 5 to the x is equal to 25.

[00:00:06]And here I give you guys this options.

[00:00:08]So go ahead and pause the video and think what the answer is and put in the comment before you watch the solution.

[00:00:15]Ready? Okay. So this is what we'll do first. I'm going to check these three options and see what the answer is if any of them. So if I plug in one to these two x's, we get 5 to the 1st + 5 to the 1st. Well, that's just five + 5 which is equal to 10. But we're trying to get 25. So a is not the answer.

[00:00:38]Continue with B. It looks promising, huh? Because we have five to the second power. However, we have another one. So that's plus 5 to the second power. This is 25 already. And then here we have another 25. Together we get 50. That's not it. So also not the answer. Okay. C.

[00:01:02]Plugging 1/2 into these two x's 5 to the 1/2 power plus 5 to the 1/2 power.

[00:01:09]Hm. How do we deal with the 1/2 power?

[00:01:13]Well, the answer for that is it's the same as the square root. This right here is the same as square root of five plus this right here is the same as square root of five.

[00:01:23]And just a quick reason why the 1/2 power is the same as a square root.

[00:01:27]Suppose you don't know what five to the 1/2 power is. We can call that K. If we square both sides, we can multiply these two exponents. Half * 2 is just one. So on the left we get five. On the right we get K squared. Well, I want to solve for K. Let me take the square roots of both sides. Technically I will put plus or minus. But five to the 1/2 power is positive. So you don't need this. So you can see K is the same as square root of five, which is the same as 5 to the 1/2 power.

[00:02:03]But anyways, here we have square<unk> of 5 plus square<unk> of 5 H. Is that 25? No. This right here is two of them, right? 2 * square<unk> of 5.

[00:02:18]This is for sure not 25. You can work this out. This is the same as square<unk> of 4 time square<unk> of 5.

[00:02:27]You get square<unk> of 20. That is not 25 because if you want to get 25 that will be square root of 25 squar which is 625.

[00:02:37]This will have been 25. But of course not that. So C is also not the answer either. Okay.

[00:02:47]Yay. We got it. It has to be D now, right? Because none of the above done.

[00:02:52]Of course, I'll show you guys how to solve this. So, check this out. All right, let me rewrite the question again. Right here, we have 5 to the x power + 5 to the x. It's equal to 25.

[00:03:04]Here we have two of them and we're adding them up. So, we just have 2 * 5 to the x.

[00:03:11]Keep in mind, do not add the exponents.

[00:03:14]5 to the x and 5 to the x, they are the same thing. when we combine them we just have two. So just adding the coefficients one and one we get a two.

[00:03:24]Next we are going to get five to the x by itself. To do so just divide both sides by two and we see we have x in the exponent. So this is an exponential equation and we have two ways to do it. One is to see if the right hand side can be written as five to some power. Yes, 25 is the same as five square but two not so much.

[00:03:51]Right? So we'll do the second way which is we use logarithm.

[00:03:56]So we will just take log with the same base which is five here. We do that both sides so that this and that can cancel and we'll just get x by itself which will be log base 5 of 25 / 2. In my opinion this is a totally okay answer.

[00:04:16]You can just leave it like that where you just have one single logarithm which is totally perfect. And of course sometimes notice we can simplify this a little bit if you want to do so. This is how when we have log base 5 of 25 / 2.

[00:04:33]Notice we have a division on the inside.

[00:04:36]So we can break it apart as a subtraction.

[00:04:41]This is log base 5 of 25 minus log base 5 of 2.

[00:04:50]This right here is actually a nice number because 25 is the same as 5^ squar. So we can write this as log base 5 of 5^ squar. But this right here is just that you cannot simplify it too much. You can use a calculator to get an approximation. But this is an irrational number. So I'm just going to leave it.

[00:05:12]But for this you can cancel this and that. You'll just get two. So you can write the answer as 2 minus log base 5 of two. And that's it. And like always, if you only have one thing inside of the log, you don't need a parenthesis. So this is totally okay.