Sharp Olympiad Mathematics | Indian | Can You Solve This One?

[00:00:01]Okay, if you're ready, let's get the value of m that will satisfy this equation here.

[00:00:12]3 ^ m + 2 = 2 ^ 2m + 2.

[00:00:23]How do we solve this problem here?

[00:00:27]You know we we don't have the same base, even though we have compared the powers.

[00:00:33]So, what we can do is that we apply one of the laws of indices.

[00:00:38]a ^ b + c, you know it's the same thing as a ^ b * the same a ^ c.

[00:00:47]Okay, so this is what we're going to have as we have 3 ^ m then multiply by 3 again ^ 2.

[00:01:00]And this is equal to 2 ^ 2m and it's multiplying 2 ^ 2. Can you see it?

[00:01:12]Okay, if you're following, see what you'll do next. 3 squared is 9, 2 squared is 4. So, we can write 3 ^ m * 9 = 2 ^ 2m * 4.

[00:01:26]And again, look at one more thing that we can do.

[00:01:31]This is 9 into 3 ^ m.

[00:01:36]Then here, you know we can do this as 2 ^ 2 then we take our m outside.

[00:01:44]Or if I want, I want to write 4 first, so I'll put 4 here into 2 ^ 2 and then I'll take the m outside.

[00:01:57]I've not changed the equation.

[00:02:01]So, this is 9 * 3 ^ M.

[00:02:05]Now, this is 4 * 4 to the power of M. If you like, you can now bring them together.

[00:02:14]So, we have power of M.

[00:02:18]So, from here now, our next target is that the terms with M, we bring them together and the constant, we bring them together.

[00:02:28]Okay. So, here is what I want us to do.

[00:02:32]I would like us to remove 9 from here. So, that means we divide 9 by itself and divide this by by 9 as well.

[00:02:43]Okay. So, on the left-hand side, we're going to have 9 to cancel 9 and we have 3 ^ M.

[00:02:55]And it's equal to 4 over 9.

[00:02:58]This is multiplying 4 ^ M.

[00:03:02]So, to bring 4 ^ M to the left, that means I will divide both sides again by 4 ^ M.

[00:03:10]Divide this by the same 4 to the power of M.

[00:03:14]And now, what do you think we're going to have?

[00:03:18]If this goes with this one, then on the left-hand side, we just have 3 ^ M divided by 4 ^ M.

[00:03:28]And that is equal to 4 over 9.

[00:03:32]Okay, that is 4 over 9. So, this is what we have. But, we can apply one of the laws of indices here.

[00:03:41]Power of M and the same power of M. So, we can combine them and let them have same power.

[00:03:48]This is coming from one of the laws.

[00:03:51]You know, if you have X to power B over Y to B.

[00:03:57]This is X over Y both combined to power B.

[00:04:03]Okay, this is interesting. So, we do the same thing to the left and we're going to have 3 over 4.

[00:04:10]This is combined to the power of M.

[00:04:14]And then we have 4 divided by 9.

[00:04:19]But then, you can see that we don't have the same powers.

[00:04:22]And we do not have the same base. So, what should we do?

[00:04:26]Take the log of both sides.

[00:04:30]Like I said, we have to take the log of both sides to have log 3 over 4 to the power of M equals log 4 divided by 9.

[00:04:46]And from here we know that one of the laws of indices um logarithm says that this power here can be brought down so that we have M log 3 over 4 to be equal to log 4 over 9.

[00:05:08]And then, the next step is that we are going to apply another law. Yes, it's possible because if you have log X and you're dividing it by Y you can write it as log X then subtract log Y.

[00:05:30]So, if we go on we're going to have that M multiplying log 3 minus log 4.

[00:05:40]And then on the right, we're going to have log 4 minus log 9.

[00:05:45]I'm applying this particular law.

[00:05:48]So, if we proceed to get the value of M now, we have to do away with this.

[00:05:53]This is log 3 minus this is log 4.

[00:05:57]Divide this by log 3 minus this by log 4.

[00:06:03]So, this will cancel this for us. So, we have our M to be log 4 minus log 9 divided by log 3 minus log 4.

[00:06:18]If the question says, "Sorry, I wrote out of sight."

[00:06:21]If the question had said that you should leave your answer in terms of log Okay, so from these points let's get our answer in approximated form.

[00:06:33]So, log 4 minus log 9 is going to give us -0.352.

[00:06:39]-0.352 Then, log 3 minus log 4 will give us -0.125.

[00:06:51]125. So, the negative will cancel itself.

[00:06:55]And let's get the approximated value of M which is going to be 0.352 divided by 1 divided by 0.125.

[00:07:08]And it's giving us 2.816.

[00:07:11]2.816.

[00:07:14]So, this is the approximated value of M.

[00:07:18]But, remember that we have not verified, right? So, we have to verify this to be sure.

[00:07:24]Let's Let's Let's try it.

[00:07:29]Okay.

[00:07:31]So, this is our um original equation.

[00:07:36]And our M from calculation is approximately 2.816.

[00:07:42]So, we're going to put this value into the equation very quickly.

[00:07:50]Okay, so this will give us three to the power of two.816 + 2 and that should be equal to two to the power of two * 2.816, right?

[00:08:07]So close it and we are going to add two here because of that. So let's take a step further.

[00:08:19]So the addition of these will give us three to the power of four.816 Then on the other hand, we have two to the power of the product of this is 5.632.

[00:08:33]So we have 5.

[00:08:35]632 and we still have + 2, right?

[00:08:39]So this is three to the power of 4.816 Then here we have two to the power of 7.6 32.

[00:08:53]Now let's use calculator to get the approximated figures for the both um sides.

[00:09:00]On the left hand side, we have 198.53, 198.53 approximately and on the right we have 198. what?

[00:09:15]. um 36.

[00:09:18]You can see that the left hand side and the right hand side are closely equal to each other, right?

[00:09:24]Yes, they are approximately equal to each other. So we are good to say that M is equal to or is approximately equal to two.816.

[00:09:35]Thank you for watching.