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

Ajouté : 2026-06-03

[00:00:00]In this video we want to solve for x given 4^ x + 4^ x + 4^ x / 6 is = to 36 From here since 4 raised to power x is being added to itself three times we can factoriize that as 4 raised ^ x * 3 / 6 is = 36.

[00:00:50]Then 3 here is 1. 3 here is 2 giving us 4^ x / 2 is equal to 36.

[00:01:04]The next thing I'm going to do at this point will be to cross multiply. So we have this * this is equal to this * this giving us 4 raised to power x * 1 is = 2 here * 36.

[00:01:25]This will imply 4^ x is = 2 * 36 here we can express as 4 * 9.

[00:01:40]Then I'm going to divide both sides by 2 * 4 which is 8.

[00:01:49]So that this here takes care of this.

[00:01:52]And then we have 4 raised ^ x / 8 is = 9.

[00:02:03]I'm going to write 4 as 2^ 2 then raised to power x and then I'm going to write 8 as 2^ 3 then = 9. Now this expression here is of the form P raised to power a raised to power N and by law of indices we can express that as B raised to power a * N.

[00:02:34]Our equation then becomes 2^ 2 * x here 2x then / 2^ 3 is equal to 9.

[00:02:52]Now this expression is of the form p raised to power a / p raised to power m and by law of indices this gives us p rais^ a - m. So we are going to have 2^ 2x - 3 is = 9.

[00:03:24]This is an exponential equation. Let us take the logarithm of both sides. So log 2 raised to power 2x - 3 is equal to log 9.

[00:03:39]The left hand side expression is of the form log p raised to power c by logarithm.

[00:03:47]This will give us c * log p. Therefore, the equation becomes 2x - 3 * log 2 is = log 9.

[00:04:12]Then I'll divide through by log two.

[00:04:20]So that this here takes care of this leaving us with 2x - 3 is = log 9 / log 2. We can also write this as 2x - 3 is = log let's express 9 as 3^ 2 then / log 2 which will then become 2x - 3 = 2 * log 3 / log 2 Now log 3 / log 2 is of the form log a / log b and by log of logarithm this will give us log a base b. So this expression here becomes log 3 b 2. Then we have 2x - 3 = 2 * log 3 base 2.

[00:05:38]Therefore 2x is now equal to pos3 + 2 log 3.

[00:05:51]I'll divide through by two on both sides.

[00:05:56]Two here takes care of this. Two cancels two here. Leaving us with x is = 3 / 2 + log 3 base 2.

[00:06:11]This will then be our final answer to this problem.

[00:06:16]Let us now check to confirm that this is correct.

[00:06:21]The problem given was 4^ x + 4^ x + 4^ x divided by 6 to give us 36.

[00:06:38]Before we go ahead and substitute for x in this equation, recall that we were able to reduce this to give us 4 raised to power x * 3 / 6 to give us 36.

[00:07:00]And now we can make our substitution for x. The value for x is 3 / 2 + log 3 2. So we put this now in this position which will then give us 4 raised to power 3 / 2 + log 3 base 2 then * 3 / 6 to give us 36.

[00:07:37]3 here is 1. 3 here is two. Then we go ahead and separate these powers using this law of indices. P^ a + n will give us p raised to power a * p^ n.

[00:07:54]Therefore we now have 4^ 3 / 2 * 4^ log 3 2 this is 1.

[00:08:07]then divided by this is 2 to give us 36.

[00:08:16]4^ 3 / 2 is written as saying<unk> 4 then^ 3 * here this is we can express this 4 as 2 ^ 2 then* log 3 2 all / 2 to give us 36 4 is 2 so we have 2^ 3 * I'll take this all the way here by log logarithm. So I have 2 power log 3^ 2 base 2 then / 2 to give us 36 2^ 3 here is 8. So we have 8 here * 2 power this will be log 3^ 2 is 9 then this 2 / 2 give us 36 2 here 1 2 2 here is 4 giving us 4 * 2 ra^ 9 to give us 36 6.

[00:09:32]Looking at this expression here, this is of the form P raised to power log M P and by law of logarithm, this will give us M. So this expression here will give us 9. Then we have 4 * 9.

[00:09:53]So give us 36.

[00:09:57]4 * 9 is 36.

[00:10:01]to give us 36. And since the left hand side balances the right hand side, that confirms that the value we got for x, which is 3 / 2 + log 3 is absolutely correct. Thanks for watching. Please like and share and also remember to subscribe to my channel and I'll see you in my next video. Bye.