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

Added: 2026-04-29

[00:00:01]In this video, let us solve for a given 4^ a + 4 power a is equal to 400.

[00:00:24]We're given 4 raised to power a + 4^ a is equal to 400.

[00:00:37]We can factoriize 4 to power a here into brackets. Here is one here is also one. Then equal to 400.

[00:00:52]This will then give us 4^ a * 1 + 1 is 2 then = 400.

[00:01:05]We can divide both sides by 2.

[00:01:10]This here takes care of this. Two here is one. Two here is 200.

[00:01:18]Then we have 4^ a is = 200.

[00:01:27]This is an exponential equation. Let us take the logarithm of both sides. So log 4 raised to power a is equal to log 200.

[00:01:42]log 4^ a on the left is of the form log m raised to power x by law of logarithm this will give us x * log m then this will then imply a * log 4 is equal to log 200.

[00:02:11]The next thing we'll do will be to eliminate logarithm from the left hand side. So we divide both sides by log four.

[00:02:25]So this takes care of this leaving us with a is equal to log 200 / log 4.

[00:02:44]Let us break down this 200.

[00:02:47]We can start with a long division.

[00:02:51]200 / 2 will give us 100.

[00:02:56]If we divide again by two, we have 50 here. And we know that 50 is 25 * 2.

[00:03:07]Therefore, 200 is equivalent to 2 raised to power 3* 25.

[00:03:22]So we replace 200 here with 2^ 3 is 8 and then 25 which is also 5^ 2 but let's start with this. This will then give us a is equal to log. So instead of 200 I'm going to write 2^ 3 * 25 or we can say 5^ 2 then divide by log 4 the expression here can be separated using this law of logarithm given log b * m by law of logarithm this will give us log b + log m then this will become log 2^ 3 + log 5^ 2. So we have a = log 2 raised to power 3 + log 5 raised to power 2.

[00:04:35]Then both divided by log 4.

[00:04:42]Next we'll separate this division to give us a is equal to log 2^ 3 then divided by log 4 then plus log 5^ 2 also divided by log 4.

[00:05:09]We can rewrite this as a = log 2 power 3 / 4 here is 2^ 2. So we have log 2 raised to power 2 plus log 5^ 2 / log 2 rais^ 2 here as well.

[00:05:41]Then we apply law of logarithm to each of these terms to give us a equal to here this will be three log 2 divided by here this will be two log 2 then plus this will be 2 log 5 / this will be 2 log 2.

[00:06:08]Now we take a look to see if anything cancels. We see two here takes care of this and then log two here cancels log two here leaving us with a = 3 / 2 + log 5 / log 2.

[00:06:33]log 5 / log 2 here is of the form log m / log p by logarithm this will give us log m base p. So this will become log 5 b 2 then we have a is = 3 / 2 + log 5 base 2.

[00:07:06]This would then be our final answer to this problem.

[00:07:10]Let us now do a quick check to confirm that this is correct. To do that, we'll need to substitute this value of a back into the given problem which was 4 raised to power a + 4^ a is = 400.

[00:07:34]We said this is 4^ a * 2 = to 400.

[00:07:41]So in place of a here let us put 3 / 2 + log 5 base 2.

[00:07:49]So we would have 4 raised ^ 3 / 2 + log 5 is 2 to give us let's say* 2 here. So all of this time 2 I'm going to put this in brackets and say time two to give us 400.

[00:08:17]We'll need to separate these powers. To do that, we use this law of indices.

[00:08:22]P^ M + N.

[00:08:27]This will give us P^ M * P R^ N.

[00:08:34]Therefore this becomes 4 raised to power 3 / 2 * 4 raised to power log 5 base 2 then* 2 to give us 400.

[00:08:54]This right here is square root of 4 raised to power 3. And we can express this four here as 2 raised to power 2 then times log 5 is 2. Now * 2 to give us 400 square root of 4 is 2. So we have 2^ 3 times by law of logarithm we can bring this back here. This will then be 2 to power log 5^ 2 base 2 then * 2 to give us 400 2^ 3 is 8 then times this becomes 2 raised to power log 5 to power 2 is 25 25 is 2 * 2 to give plus 400.

[00:10:02]We can do 8 here * 2 here. This will give us 16.

[00:10:08]This remains unchanged.

[00:10:10]So 2 raised to power log 25 base 2 then equal to 400.

[00:10:20]Now this expression here is of the form p raised to power log m by law of logarithm this will give us m.

[00:10:32]So this expression here will give us 25.

[00:10:37]This will then imply 16 * 25 should give us 400.

[00:10:50]16 here is 4 * 4.

[00:10:54]Then * 25 to give us 400.

[00:11:00]4 * 25 here is 100.

[00:11:04]And then 100 * 4 to give us 400.

[00:11:11]4 * 100 is definitely 400. So we have 400 here is equal to 400 here. Now since the left hand side balances the right hand side it confirms that the value we got for a which is 3 / 2 + log 5 base 2 is absolutely correct. Thanks for watching. If you have enjoyed this video, please give it a thumbs up and share. And also remember to subscribe to my channel. And I will see you in my next video. Bye.