Math Olympiad | Exponential Equation | Germany | Can you solve this?

[00:00:00]Hi everyone.

[00:00:03]Let's provide the solution to this one here.

[00:00:09]4 to the power of x + 1 = 40.

[00:00:17]So, what we are going to do is to um work on this using one of the laws of indices that we know.

[00:00:26]This will be 4 to the power x * 4 to the power of what? To the power of 1. And this is equal to 40.

[00:00:38]Remember, if you want to pick one of the bases from the left-hand side, you are going to have um to pick four and then add the powers.

[00:00:49]So, from here now, this is the same as having 4 * 4 to the power of x, which is equal to 40.

[00:00:58]So, from here, we have to remove this four by dividing both sides by the same four there.

[00:01:07]And here, we do the same so that this will remove this four of us.

[00:01:12]And now, we have 4 to the power of x to be equal to 40 / 4, and that is going to be 10.

[00:01:23]Guess what I will do.

[00:01:25]Since we are not having the same base, just take the log. And you have log 4 to the power x to be equal to the log of what? 10.

[00:01:36]The next point, I know that log 10 is the same as um one, but I prefer working with that log 10 so that I can bring the power down to give me x log 4, which will be equal to log uh 10.

[00:01:53]Divide this by log 4 so that x will be free, and the same thing will happen here.

[00:02:00]These will cancel out.

[00:02:01]But, from the right, we cannot cancel anything. So, we write our X to be equal to log 4 log 10 rather divided by log 4.

[00:02:18]Now, from these points From these points, what do we do? We can apply change of base.

[00:02:27]Do you know about change of base that says that if you have B that is raised to the power of A to the same base of B. Like, everything here is equal to A.

[00:02:41]Do you see that? So, if that is true, then what we have right now can be written like this, X equals log 10 now to the base of Okay, I think it's a different law that I explained. It's a different law that I explained. I'll use that law later.

[00:03:00]Now, [snorts] this law here says that if you have log M divided by log B that everything you have here, let me try to show it. Everything you have here is the same thing as log M to base B.

[00:03:17]See that?

[00:03:18]So, this B now becomes the base to M.

[00:03:21]So, the same thing is what will happen so that 4 here becomes the base to the 10.

[00:03:27]As simple as that.

[00:03:29]So, at this point, we have our value of X in terms of log.

[00:03:34]Now, we will not stop here. Let's go very quickly to the original equation.

[00:03:39]So, this is the original equation and our value of X is log 10 to base 4.

[00:03:49]Log 10 to base 4, right?

[00:03:52]Okay, let me write this better.

[00:03:56]We have log log 10 to base 4 is our value of X.

[00:04:05]So, in place of X there now, put this value. So, we have 4 to the power of log 10 to base 4 + 1.

[00:04:18]This is one, right?

[00:04:20]And then, if we continue, we are looking for what this will give us. If it does not give us 40, then we are not correct.

[00:04:27]So, this is the same thing as 4 to the power of log what? Log 10 to base 4 multiplied by 4 to the power of 1.

[00:04:36]From one of the laws of indices, so that I explained this law before.

[00:04:42]Log to base 4 and this 4 will cancel out. So, here we have 10.

[00:04:47]Multiplied by 4 to power 1, which is the same as 4.

[00:04:51]10 * 4 is 40.

[00:04:54]So, this means that we are very correct because we were looking for the 40.

[00:04:58]So, we are saying that our X to be what?

[00:05:01]Two.

[00:05:03]No, our X to be equal to log 10 to base 4 satisfies this very equation.

[00:05:13]4 to the power X + 1 = 40.

[00:05:17]Thank you for watching.