Germany | Can you solve this ? | Maths Olympiad

[00:00:00]Hello, you are welcome to solve this math problem of h a is equal to 8,000 to find the value of a from this equation.

[00:00:12]Solution from this problem here which is 8^ a is equal to 8,000.

[00:00:26]In the first step, we'll apply log in both sides. So it will be log of 8^ a is equal to log of 8,000.

[00:00:39]Then this power of a will move to the base. So it will be a * log of 8 is equal to log of 8,000.

[00:00:50]Then we'll divide by log of 8 in both sides. So here over log of 8 and here over log of 8. Then this log of 8 we cancel with log of 8. Then it will be a is equal to log of 8,000 over log of 8.

[00:01:16]Then in the next step it will be a is equal to log of 8,000 12 8,000 is same as 8 * 1,000 then bracket over log of 8.

[00:01:35]Then here it will be a is equal to log of 8 * 1,000. This part here is in the form of log of a * bracket which is equal to log of a plus log of b. So we'll apply this rule here into this form. Then it will be log of 8. So here log of 8 + log of 1,000 then divide by log of 8. So here it will be over log of 8. Then in the next step it will be a is equal to will be this over this. So log of 8 over log of 8 plus this over this log of 1,000 over log of 8.

[00:02:40]Then here it will be a is equal to log of 8 / log of 8 is 1. So it will be 1 + log of 1,000 over log of 8.

[00:02:58]Then here it will be a is equal to 1 + log of 1,000 is same as 10^ of 3. Then over log of 8 is same as log of 2^ 3.

[00:03:16]Then here it will be a is = 1 + here this power three will move to the base.

[00:03:23]So it will be 3 log of 10 over also this three will move to the base. So it will be 3 log of 2.

[00:03:35]Then this three will cancel this three.

[00:03:38]Then in the next step it will be a is equal to 1 + to be this over this. So log of 10 over log of 2.

[00:03:51]Then here it will be a is equal to 1 + log of 10 to 2 10 into prime number it is 5 * 2 bracket over this log of 2.

[00:04:11]Then here it will be a is = 1 + here log of 5 * 2 is in this form. Then we change into this form. So here it will be log of 5 plus log of 2 then over this log of 2. Then in the next step it will be a is equal to 1 +ide by log of two in this part will be this over this. So log of 5 over log of 2 plus this over this log of 2 over log of 2.

[00:05:02]Then here it will be a is = 1 + into your log of 2 / log of 2 it is 1. So it will be + 1 + this here log of 5 over log of 2. Then here to be a is equal to 1 + 1 it is 2 plus log of 5 / log of 2 is same as log of 5 base of two. So this is the final answer for the value of a into this problem. Now in the next step let's check this answer if it is correct. So to check from our problem which is 8^ of a is equal to 8,000.

[00:05:55]So here we substitute our a it is this here. So it will be 8^ of a which is 2 + log of 5 base of 2 is it = 8,000.

[00:06:13]Then from this part here, this part is in the form of the rule which is a power of m + n which is equal to a power of m * a^ of n. So we apply this rule here into this form. Then it will be 8^ 2. So here 8^ of 2 * 8^ log of 5 base of 2. So here 8^ of log of 5 base of 2 is it = 8,000.

[00:06:52]Then from here it will be 8^ of 2 * here 8 is same 8 is same as 2^ of 3 because we make base of two because here we have two then times this here which is log of 5 base of two this then is it equal to 8,000 then here it will be 8² * here this three will move to the power. So it will be 2 log of 5^ of 3 base of 2 is it = to 8,000.

[00:07:37]Then here it will be 8^ of 2 * 2^ of log of 3 5^ 3 is 125 5 * 5 * 5 then base of 2 is it equal to 8,000.

[00:07:59]Then here it will be 8² * here 2 log of 125 base of 2. This part here is in the form of a power of log of b base of a which is equal to b. So we'll apply this rule here. Then here 2^ log of 125 base of 2. This here is same as 125. Then is it equal to 8,000?

[00:08:34]Then from here 8² 8² same as 8 * 8 then times this here 125 is it = 8,000.

[00:08:50]Then here we multiply this here. So it will be 8 * 8 * 5 is 40 go 4 8 * 2 is 16 + 4 it is 20 goes 2 8 * 1 is 8 + 2 it is 8 + 8 8 * 1 is 8 + 2 it is 10 then is it equal to this 8,000 then 8 * 1,000 it is 8,000 000 is equal to 8,000.

[00:09:29]So into here left hand side is equal to right hand side. So it is true for the value of a it is 2 + log of 5 base of two. Thank you for watching. Don't forget these tips to miss out. Subscribe to my channel and see you in the next video. Bye-bye.