Solving a 'Harvard' University entrance exam question

Added: 2026-05-10

[00:00:00]Hello everyone, you are welcome. Today we have a very interesting exponential math problem.

[00:00:07]Here is 5 raised to power x divided by 25 is equal to 50. So here we will try to find out the value of x.

[00:00:15]So let's start our solutions. First of all, here we will take this number to the right hand side.

[00:00:20]So this expression, this equation will become this is 5 raised to power x is equal to this will become 25 * 50.

[00:00:31]Now as the variable in the power, so here we will take common log on both sides to find out the value of x.

[00:00:38]So let us take common log on both sides.

[00:00:41]So this is log of 5 raised to power x is equal to log of 25 * 50.

[00:00:52]And in both sides of this equation, here we will use different log math properties. So here we will use log of a raised to power x it is x * log of a.

[00:01:02]So this left hand side will become this will become x * log of 5 is equal to and here we will apply log of a * b which is equal to log of a + log of b.

[00:01:17]So this will become log of 25 plus log of 50.

[00:01:26]Now as we have to find out the value of x, so here we will try to eliminate this number from the left side. So here we will divide both sides by log of 5.

[00:01:36]We will also divide log of 5 here.

[00:01:40]And the left hand side, this log of 5 and this log of 5 will be cancelled.

[00:01:44]So this will become only x is equal to and here we can write this right hand side as this will become log of 25 or 25 can be written as 5 squared divided by log of 5 plus and this will become log of 50 but here we can write this 50 as this is 25 * 2 divided by log of 5 Here we'll apply log of a raised to power x which is x times log of a so this will become x is equal to this will become 2 times log of 5 divided by log of 5 plus and this will become log of 25 plus log of 2 divided by log of 5 Here we'll cancel log of 5 log of 5 so this will become x is equal to this is just 2 plus Here we can write this expression as this will become log of 25 but 25 is 5 squared divided by log of 5 plus log of 2 divided by log of 5 And again we will apply the exponential log property and we will take this two to the front.

[00:03:17]This will become x is equal to this is 2 plus this will become 2 times log of 5 divided by log of 5 plus This expression will be the same log of 2 divided by log of 5 And we'll cancel log of 5 and log of 5 so this will become just 2 and here in this expression we will use change of base logarithm method property.

[00:03:46]So in this expression we will use this one logarithm method property. Here we can log of m / log of n as log of m to base n.

[00:04:03]So, using this log property here, this expression will become This is x = This is 2.

[00:04:11]So, this is 2 and this will also 2.

[00:04:15]Plus and this expression will become log of 2 with base 5.

[00:04:22]Let us sum of these two numbers. Here, 2 + 2 is simply 4.

[00:04:26]So, the final value of x will become x = 4 + log of 2 with base 5.

[00:04:34]So, finally, this is the final value of x in terms of log.

[00:04:40]Here, we will try to verify this value of x. Let us this value of x verify this one interesting exponential equation or not.

[00:04:48]So, we'll verify this value here.

[00:04:51]To verify this value here, we will use and write this equation again.

[00:04:56]So, here the equation is simply 5 raised to power x / 25 is equal to 50.

[00:05:05]Now, first we will take this number to the right hand side.

[00:05:08]This will become 5 raised to power x is equal to 25 * 50.

[00:05:15]Here, we'll substitute the value of x.

[00:05:18]So, this will become this is 5 raised to power x is simply 4 + log of 2 with base 5.

[00:05:27]is equal to Let's multiply these two numbers. So, here 25 * 0 is 0.

[00:05:32]25 * 5 is 5 * 5 is 25.

[00:05:36]5 * 2 is 10. 10 + 2 is 12.

[00:05:39]So, this is 1,250.

[00:05:41]So, let's simplify this left hand side.

[00:05:43]So, here in the left hand side, we will use an exponential math property.

[00:05:47]So, here we will use this one exponential math property. A raised to power m plus n that can be written as A raised to power m times A raised to power n.

[00:05:59]So, using this exponential math property, this left-hand side will become This will become just is just 5 raised to power 4 times 5 raised to power log of 2 with base 5 is equal to 1,200 and 50.

[00:06:19]And in this one number, we will use another log math property.

[00:06:23]Here we will use this one log property.

[00:06:25]We can write A raised to power log of B with the same base A is equal to B.

[00:06:33]>> [clears throat] >> So, using this log property here, this whole expression will be This will become 2.

[00:06:39]So, this will become here 5 raised to power 4. It is simply 5 * 5 is 25. 25 * 5 is 125. And 125 * 5 which is about 625 times and this is just 2.

[00:06:54]is equal to 1,200 50.

[00:06:58]Let's multiply these two numbers here. 2 * 5 is 10.

[00:07:02]2 * 2 is 4. 4 + 1 is 5. 2 * 6 is 12.

[00:07:06]So, this is 1,250 is equal to 1,250.

[00:07:11]Both sides are equal. So, it's means that this value of x is the correct and exact value of x in this interesting exponential math problem.