Solve for a in this nice Algebra equation | Math Olympiad Mathematics

追加: 2026-05-25

[00:00:01]In this video, let us find the value of a given 5 to power a + 5 power a is equal to 100.

[00:00:25]We are given 5 raised to power a plus 5 raised to power a is equal to 100.

[00:00:37]This is same thing as saying 5 raised to power a into bracket 1 + 1 is equal to 100.

[00:00:52]This will give us 5 to power a * 1 + 1 here is 2 equal to 100.

[00:01:04]I'm going to divide both sides by two.

[00:01:10]So two here cancels two here. Two here is 1. 2 in 100 is 15.

[00:01:18]giving us 5 raised to power a is equal to 50.

[00:01:28]We can also write this as 5 to power a is equal to 50 here is 25 * 2.

[00:01:40]Let us divide both sides by 25.

[00:01:48]So that this here takes care of this leaving us with 5 to power a / 25 is equal to 2.

[00:02:03]This will imply 5 to power a / let us express 25 as 5^ 2 then equal to 2 5 to power a / 5^ 2 is of the form p raised to power m / p raised to power n and by law of indices this will give us p raised to power m - n. Then we express this using this as 5^ a - 2 giving us 5^ a - 2 is equal to 2.

[00:02:52]This is an exponential equation. Let us take the logarithm of both sides. So we have log 5 raised to power a minus 2 is equal to log 2.

[00:03:08]This left hand side expression is of the form log x raised to power p. By law of logarithm this will give us p * log x.

[00:03:23]In this case our p is a minus 2. So rewriting this in this form will give us a minus 2 time log 5 is equal to log 2.

[00:03:43]Now I'll eliminate log five from the left hand side by dividing through on both sides by log five.

[00:03:55]So this here takes care of this and we are left with a a - 2 is equal to log 2 / log 5.

[00:04:12]log 2 / log 5 is of the form log x / log p by love logarithm. This will give us log x base p.

[00:04:29]So that a minus 2 becomes log 2 base 5.

[00:04:39]Then we can transfer this minus2 to the right hand side to give us a = -2 becomes pos2.

[00:04:49]Then plus this log 2 base 5 which would then be our final answer to this problem. Let us now do a quick check to confirm that this is correct.

[00:05:07]To check we need to substitute this new value of a back into the given problem which was 5^ a + 5 to power a is equal to 100.

[00:05:23]So we can individually substitute for a here or we can simplify this as we did earlier in the video into 5 5^ a * 2 is = 100 and then we just substitute for a here which we then imply five raised to power a is 2 plus log 2 base 5. Then all of that time 2 to give us 100.

[00:06:05]The next thing we'll do from this stage will be to separate these powers. To do that we apply law of indices. Given p raised to power x + y.

[00:06:18]This will give us p raised to power x * p raised to power y.

[00:06:26]So this expression becomes 5 raised to power 2 * 5 raised to power log 2 base 5 then * 2 to give us 100.

[00:06:43]5 raised to power 2 is 25.

[00:06:47]Then times these two here. Then times all of this 5 to power log 2 base 5 to give us 100.

[00:07:01]This here will give us 50.

[00:07:05]Then times here 5 raised to power log 2 5 to give us 100.

[00:07:15]Let us work on this expression here. 5 to power log 2 5 is of the form a raised to power log x base a which will give us x by logarithm.

[00:07:31]Therefore this expression will reduce to two.

[00:07:35]Then we have 50 * 2 to give us 100.

[00:07:47]50 * 2 is 100.

[00:07:50]So 100 is equal to 100. Therefore the left hand side balances the right hand side. And that confirms that this value we got for a is absolutely correct.

[00:08:03]Thanks for watching. If you have enjoyed this video, please give it a thumbs up and remember to subscribe to my channel if you have not done so already. And I'll see you in my next video. Bye.