Löse die Gleichung OHNE Taschenrechner! – Exponentialgleichungen

Hello everyone! Today I have brought you this equation that we are going to solve without a calculator. In this video I would like to show you two different ways to get to this point. One way without the logarithm and one way with the logarithm. Why logarithm at all? Well, if we look at our equation, we see that the x is in the exponents. We have x here and you normally need the logarithm to get this x out of there. But I told you there is a way without the logarithm and let's take a look at it together now. But this is only possible if the equation is set up as nicely as in this case, for example, that the basis of this 5 something on the other side basically also occurs again. Not quite as it says here yet, but in the 25 the 5 is in there. And that's the first thing we do, that we write the 25 so that we can see better that there is a 5 in there. So we leave the left side as it is, it fits so far. But on the right we write this 25 like this in 5 raised to 2. And then we see nicely, aha, there is a 5 as a basis, there is a 5 as a basis. It is very suitable that you maybe compare this with each other. But not yet as it says here, because there is a break here. We still can't compare both sides well. We have to somehow rewrite the break and we can do that too, we leave the left side as it is. But you can rewrite such a break, such a 1 divided by a power by simply making the raised number negative. So we write 5 raised and now we don't write 2, but minus 2. That's exactly the same, just a different way of writing. So if you have negative raised numbers, something like x raised to minus 7, then that's just another way to write down a break. A 1 through, there is always 1 through and then there is this power here again, but instead of the negative raised number you make a positive raised number out of it. And that's exactly what we used here. And now we can compare both sides really well, because on the left side there is a 5 as a basis and also on the right side there is a 5 as a basis. If both sides should be the same, then the exponents have to be the same, so that these two sides can be the same at all. That means, if we compare them, then 1 minus x has to be exactly the same as the minus 2 on the other side. Only in this way can both sides be the same. And that's a new equation, it's cute. We can solve it for x by bringing 1 to the right with minus 1, then there is minus x here, don't forget the minus. Minus 2 minus 1 is minus 3 and so that x is alone, the minus has to be removed. So we multiply with minus 1, then the minus disappears. And on the other side, minus 3 times minus 1 is 3. And we have found our solution to our equation without the logarithm. But also with logarithm, the whole thing works. But also without a calculator. We have to remember that we are not allowed to calculate anything here. How does it look? Logarithm, let's just hit it, or let's think for a moment about what basis we will use. Since we are not allowed to use a calculator, I would use the basis that is also present here, namely the basis 5. Then the calculations are much easier. That means, on the left side we turn the logarithm to the basis 5 of everything that was on the left side. We just write that down. And on the right we turn the logarithm to the basis 5 of everything that was there. And then it goes into the logarithm laws. Because we have to see how we get the whole thing transformed so that we don't have to calculate much here. We are allowed to subtract these exponents from the logarithm. That's what a logarithm law says. And we turn that around here. So the 1 minus x comes out. And because this is now multiplied here, we have to make a bracket around it because there is a difference here. But then the rest comes, i.e. the logarithm to the basis 5 of what is left here, only of the 5. On the right side, there are only numbers. You could actually calculate that with a calculator without any problems. But we are not allowed to. That's why I would say we turn that around in the logarithm just like before. So that the logarithm to the basis 5 is there. And we turn this 1 25th around so that the 5 is also seen as the basis. We saw that in the first place, that it is nothing more than 5 raised to the power of minus 2. Because it is always good when the basis and the basis in the logarithm match. We see that right away, or we already see that at this point here. Because the result of this is simply 1. So if the basis is the same as what is in here, then 1 comes out. And that's why it just says times 1, which you can omit. And then it says on the left side only 1 minus x. On the right side, we still have this high minus 2 here. But we draw that in front of the logarithm just like before. So the minus 2 comes to the front and the rest comes to the back. But it can't shock us anymore, because that's exactly what we already had here. We know that this is 1, so times 1 is here and then we can also omit the times 1. Do you recognize this equation? Do you remember it? We just saw it. We solved it in such a way that we ultimately got x equals 3. And with that we have also successfully completed the second way. You are welcome to write in the comments which way you liked better. Otherwise, have a wonderful day and we'll see you in the next video. Take care!