Interesting Logarithmic Questions

[00:00:00]Hello everyone. Welcome to my YouTube channel. So here today we are going to solve this uh equation.

[00:00:06]So what are we going to do?

[00:00:08]So the first thing is to split this.

[00:00:12]So this according to the rules of indices, if they are dividing, the rules of logs, it means to say they are going to subtract each other like this. So we are going to have this is the same as the log base eight, then it will be X. Now since two is dividing, it will be minus the log of eight over two like that should be equal to we have the log of uh eight and like that over the log of uh base eight of two.

[00:00:46]So that means to say immediately you do this. As you can see, this one and this one are the same. So this one and this one are the same.

[00:00:56]This one and this one are the same.

[00:00:59]Therefore, you can let them be equal to a common letter. So you are going to say let your A be equal to the log of eight X base eight and B be equal to the log of uh we have eight over two base eight like that.

[00:01:19]So with that said, we are going to replace these values. So where you see that you are going to say A minus we have our B should be equal to this makes our problem to be more simple. So we are going to have the log of uh so where there is the log of eight of X, it's A over this one is B like that.

[00:01:40]So with that said, from here we are going to cross multiply. So B multiplies everything, so it will be B open brackets A minus B like that should be equal to A like that.

[00:01:54]Opening the brackets, we are going to have a * b, it will be ab.

[00:02:00]b * -b, it will be -b ^ 2 should be equal to we have our a like that.

[00:02:07]From here, what we are going to do is since this b is squared, let's take it here. So, it will be ab, then the a will come this side, ab minus our a should be equal to we are going to have our b squared like that.

[00:02:25]Then, we are going to have our a is common here, so we are going to factorize it. So, we have a is common.

[00:02:35]Then, we are going to have our b minus one should be equal to b squared.

[00:02:41]When you reach this stage, when you have simplified it like this, you can now replace.

[00:02:47]So, you can now replace.

[00:02:49]So, what do we have?

[00:02:51]So, we have the log of this value the true number here. So, we are going to say our a, let me just bring it closer.

[00:03:02]So, we are going to have the a, it will be the log of eight of x like that.

[00:03:14]This is multiplying b which is the log of base eight of two minus one should be equal to where there is this we are going to have the log of eight of base two like that.

[00:03:30]So, this is what we have.

[00:03:33]So, this is squared like that.

[00:03:36]So, as you can see, the reason why we left b minus one, b minus one was okay because b is a can be expressed as a pure number because it is made up of numbers. We have eight and two.

[00:03:49]In which case, even here eight and two.

[00:03:52]So, that means to say this and this can be subtracted. So, that is better. It makes the equation to be simple. So, we have the log of 8 of x. Then, we have So, now here, we just have to be smart.

[00:04:06]2 You know to say 2 is equal to the cube roots of 8 is equal to 2. So, that means to say 2 is equal to 8 to the power 1 over 3. So, cube root is the same as 1 over 3. So, we can say that another bracket, it will be it will be log of 8 8 to the power 1 over 3 minus 1 like that. So, the reason we have introduced 8 is if we have the same base, we can easily cancel it. So, we have should be equal to We have even here, it will be the log of 8 8 to the power 1 over 3.

[00:04:53]Then, we're going to put our square there.

[00:04:56]So, what we are going to have is what we are going to have is log of our 8 like that. Then, we have this 1/3 comes here. We're going to have 1 over 3 the log of 8 base 8 minus 1 like that should be equal to We have the same thing here should be equal to We have the 1 over 3.

[00:05:25]Then, we have the log of 8 base 8 like that. Then, we have a squared there. So, what we are going to have as our value there will be equal to We have the log of um our 8 our x base 8. Then, we have This if you have the log of the same base, this is equal to 1 as long as this and this are the same. It means it is equal to one. So, we are going to have 1/3. Then we have * 1 this one. Then this should be equal to So, same here. It It will be equal to 1/3 because this times this is one. So, we have this. Then we have squared like that.

[00:06:11]So, from there what we are going to have is uh we have the log of that then times we have 1/3.

[00:06:21]So, we can take So, we can even forget about one. It doesn't change anything in that case. So, we have 1/3 - 1. It will be Three is the common denominator. So, we have 3 / 3 is 1 * 1. That's 1. - 3 / 1 is 3 * 1 is -3 like that. Then we have should be equal to 1/3 squared is 1/9 like that. So, we are going to get our log of eight like that.

[00:06:57]1 - 2 is -2/3 should be equal to 1/9 like that.

[00:07:04]With that said, what we are going now to to get is uh we know to say Let's just get rid of uh uh this so that this whole thing can be left alone.

[00:07:19]So, we can uh do this.

[00:07:23]So, we have times there.

[00:07:26]Let's move this a bit.

[00:07:28]Should be equal to 1/9. So, we can multiply the multiplicative inverse which is a 3/ -2 times a 3 over -2. So, this and this cancels. This and this cancels. So, we are going to have log of x base 8 should be equal to this times this. This cancels. This We are going to have three. So, we have negative. We can just throw the negative up. Negative one over the six like that.

[00:08:00]Then, what we are going to have is we are going to say this goes there. So, it will be It will become the base. So, it will be x is equal to 8 to the power -1/6 like that. So, this is x is equal to We are going to have x in terms of two is the same as two to the power of three like that to the power -1/6 like that. So, x is equal to two to the power of this and this We are going to have -3/6, which will give us x is equal to two to the power negative This and this cancels. So, we are going to have two there, one like that. So, x is the same as one over We have a two to the power of one over two, which will give us x is equal to one over the square root of our two like that. So, this is uh our answer for that problem. Thank you very much for watching. Hope you would like, subscribe, and share for more.

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