Germany | Solve This Olympiad Equation In 10 Seconds ( No Calculator!)

Added: 2026-05-06

[00:00:00]You're welcome to solve this nice exponential equation which is 4 ^ a + 4 ^ this is equal to 100. What is the value of a? Now let's provide a solution from here.

[00:00:15]So we have that 4 to the^ of a this is common. Let's factor out 4 the^ a. So that's 4^ a / 4 the^ this is 1 + 4 ^ a / 4 the^ a this is 1 this is equal to 100.

[00:00:34]So this is 4 raised to the power of a * 1 + 1 this is 2 and this is equal to 100.

[00:00:44]So let's divide on both sides by two.

[00:00:48]So that now 4 ^ of a this is equal to 100 / 2 and this is equal to 50.

[00:01:00]To solve for the value of a let's introduce logarithm on both sides we have log 4 ^ of a. This is equal to log 50.

[00:01:14]We have that log 4 to the^ of a this is in the form of log a raised to the power of b which we can express as b ro a.

[00:01:27]Applying this power property then we have log 4 to the^ of a becomes a rogue 4. This is equal to log 50.

[00:01:40]Now let's divide on both sides by log 4.

[00:01:45]Here we have log 4.

[00:01:49]So if you simplify here we have that a is equal to log 50 / log 4.

[00:02:03]The next step is that we can express 50.

[00:02:07]This is the same thing as 25 multiplying by by two.

[00:02:14]So this means that a is equal to log 25 multiplied by 2 then divided by log log 4.

[00:02:29]We have that log 25 * 2. This is in the form of log a * b which we can express as log a + log b.

[00:02:43]Applying this logarithm property we have a is equal to log 25 / log 4 then plus log 2 / log 4.

[00:03:02]So we have that a is equal to we can express log 25. This is the same thing as log 5 raised to ^ 2 / log 4 which is log 2 ^ of 2 then plus log 2 / log 4 which is log 2 raised to the power of 2.

[00:03:26]So let's apply the power of property here. So that a is equal to log 2 5 ^ 2 becomes 2 log 5 / 2 rogue 2 then plus log 2 / 2.

[00:03:48]So let's simplify here. 2 and two simplifies. Then we have log two and log two here simplifies. So we have that a is equal to a half plus 5 divided by log 2 log 5 / log 2. This is in the form of log a / rog b which we can express as rogue a to bit b.

[00:04:21]So that now we can express the value of a equal to a half plus log 5 to base 2.

[00:04:31]So this is the value of a.

[00:04:35]So let's check if this value of a here satisfies the equation.

[00:04:42]Now let's verify that this value of a if this satisfies the equation. If you recall, we enter that 4 to the^ of a + 4 to the^ of a. This should give us a value of 100.

[00:05:02]So 4 to the^ a is common here. Let's factor out so that a we have 1 + 1. This should give us a value of 100.

[00:05:15]So this is 4 to the^ of a multip this should give us a value of 100.

[00:05:24]So let's substitute the value of a. We have two raised to the power of a half plus row 5 to base 2.

[00:05:36]Everything here multiplying by two this should give us a value of 100.

[00:05:43]Let's express 4 as 2 ^ of 2. So this is 2 raised to the power of 2 multip which is half + log 5 to base 2 everything here multiplying by two this should give us a value of 100.

[00:06:08]So this means that we have 2 ^ of a half * 2 here this simplifies. So we have 1 + 2 * 5 to base 2. This means we have 2 row 5 to base 2 multiplying by 2 this should give us a value of 100.

[00:06:31]Now since 2 is a power we can express this as 2 ^ 1 + log 5 ^ of 2 2 base 2 multiplying by two this should give us a value of 100 we have that 2 ^ 1 + 5 ^ 2 to base 2.

[00:06:54]This is in the form of a ^ n + m which we can express as a ^ n * a ^ of m. So applying this property applying this property we have 2 ^ 1 * 2 raised ^ of 5^ squar to base 2 everything here multip by 2 this is supposed to give us a value of 100.

[00:07:25]So this means that 2 ^ 5 to the^ 2 to base 2 this is in the form of a to the power of log m to base a this should give us a value of m. Applying this property then we have 2 multiplying by 2 ^ 5 to base 2.

[00:07:46]This will give us a value of 5 squared.

[00:07:50]Then multiplying by two this should give us a value of 100.

[00:07:58]So this means that this is 2 * 5^ 2 which is 25. Then multiplying by 2 this should give us a value of 100. 25 * 2 this is 50 which is multiplying by 2 this should give us a value of 100. So 50 * 2 this is 100 which is equal to 100. So the left add side is equal to the right add side. And this affirms that the value of a which is equal to a half plus 5 to base 2 actually satisfies the equation. So kindly follow the steps like this video and subscribe. See you in the next video.