Germany | Can you solve this ? | Maths Olympiad

Añadido: 2026-06-01

[00:00:00]Hello. You're welcome to solve this math problem, which is 4 ^ a + 4 ^ a is equal to 100.

[00:00:09]To find the value of a from this equation.

[00:00:13]Now, solution.

[00:00:19]From this our problem here, 4 ^ a it is common. So, we'll take 4 ^ a outside the bracket. Then 4 ^ a / 4 ^ a it is 1 + 4 ^ a / 4 ^ a it is 1. Then bracket is equal to 100.

[00:00:39]Then into here it will be 4 ^ a * 1 + 1 it is 2 is equal to 100.

[00:00:48]Then into here we will divide by 2 in both sides.

[00:00:53]So, this two and this two will cancel.

[00:00:56]Then it will be 4 ^ a is equal to 100 / 2 it is 50.

[00:01:04]Then into here we'll apply log in both sides.

[00:01:07]So, it will be log of this 4 ^ a is equal to log of 50.

[00:01:16]Then this ^ a will move to the base. So, it will be a * log of 4 is equal to log of 50.

[00:01:27]Then into here we'll divide by log of 4 in both sides. So, here over log of 4 and to this side over log of 4.

[00:01:37]So, this and this will cancel. Then it will be a is equal to log of 50 over log of 4.

[00:01:49]Then into here it will be a is equal to log of 50. 50 into prime numbers from 50 here factors of 50 here it is divisible by two it is 25 then divide by five it is five then divide by five it is one so 50 it is two times five times five which is five squared >> [snorts] >> then over log of four it will be log of four into prime number four is two times two which is two squared then into here it will be a is equal to from log of two times five squared this part here is in the form of log of a times b which is equal to log of a plus log of b so we'll apply this form here then it will be log of two so here log of two plus log of five squared so here log of five squared then over this log of two squared then into here it will be a is equal to log of two plus log of two squared this square will move to the base so it will be two log of five then over log of two squared this square will move to the base so it will be two log of two then in the next step it will be a is equal to we divide by two log of two in this part and this part so it will be this over this so here log of two over two log of two then plus this over this two log of five over two log of two then into here it will be a is equal to log of 2 / log of 2, it is 1. So, it will be 1 / 2 here 1 / 2 plus here this two will cancel this two. So, it will be this over this log of 5 over log of 2.

[00:04:09]Then into here, it will be A is equal to 1 / 2 plus log of 5 / log of 2 is same as log of 5 base of 2.

[00:04:22]So, this is the final answer for the value of A into this our problem.

[00:04:29]Now, in the next step, let's check this answer if it is correct.

[00:04:33]So, to check for my problem, which is 4 ^ A + 4 ^ A is equal to 100.

[00:04:43]Now, into here 4 ^ A + 4 ^ A, there are two. So, it will be is same as 4 ^ A * 2 is equal to 100.

[00:04:54]Now, into power of A, we check we substitute this our answer here. So, it will be 4 ^ A substitute this 1/2 + log of 5 base of 2.

[00:05:04]So, here 1/2 plus log of 5 base of 2 then times this 2 is it equal to 100?

[00:05:17]Then into here this here 4 4 ^ 1/2 + log of 5 base of 2, this here is in the form of A ^ M + N. Then we change into A ^ M * A ^ N. So, we'll apply this form here.

[00:05:37]Then here to be 4 ^ 1/2 then times 4 ^ log of 5 base of 2. So, 4 ^ of of 5 base of 2 then times this two is it equal to 100?

[00:05:54]Then from 4 power of half 4 is same as 2 squared. Then bracket this power of a half times here 4 power of log of 5 base of 2 4 is same as 2 squared. Then bracket this power here of log of 5 base of 2 then times this two is it equal to 100?

[00:06:21]Then here this two will cancel this two so to be 2 power of 1 it is two times it will be two.

[00:06:28]Then this two here will take to the power so it will be log of 5 squared base of two then times this two is it equal to 100?

[00:06:42]Then from 2 * 2 it is 4 times it will be 2 power of log of 5 squared it is 25 base of 2 is it equal to 100?

[00:06:57]>> [snorts] >> Then here it will be 4 times here 2 power of log of 25 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.

[00:07:11]So we'll apply this rule here.

[00:07:14]Then 2 power of log of 25 base of 2 this here is same as 25 so it will be times 25 is it equal to 100?

[00:07:25]Now now from 4 times 25 it is 100 is equal to 100. So left side and right side are equal then it is true that a is equal to a half plus log of 5 base of 2. The answer in the simplest form. This is a correct solution.

[00:07:47]If you have another method, comment in the comment section or you have your solution so as to let us know.

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[00:08:09]Bye-bye.