The Viral Math Olympiad That Stumped Many! | Can You Solve It?

[00:00:01]Hello, you're welcome. I want to solve this nice exponential problem to find the value of x here.

[00:00:07]Solution from here.

[00:00:10]And what we have which is 9 raised to the power x times 9 raised to the power x equals to 72.

[00:00:20]First step here, this follows what we have. A times A which is A squared. Also, same thing multiplying can write it as 9 raised to the power x squared equals to 72 on this side.

[00:00:37]That is From here, this follows the law of indices. When we have A raised to the power M raised to the power N is the same thing as A raised to the power M N.

[00:00:51]That is this power multiplies, we have 9 raised to the power 2x equals to 72 here.

[00:01:02]Next step, we take the log on both side here. We have log 9 raised to the power 2x equals to log 72 from here.

[00:01:16]And this follows the power law of logarithm. When we have log M raised to the power P is the same thing as P log M.

[00:01:27]Then here, this becomes 2x log 9 equals to log 72 on this side.

[00:01:40]Next step, we divide both side by log 9.

[00:01:43]Divide this side by log 9.

[00:01:46]Also, divide this side by log 9.

[00:01:51]That is here, log 9 cancels with each other here.

[00:01:55]And we have equals to log 72 over log 9.

[00:02:09]We can rewrite 72 as 9 * 8.

[00:02:15]And what we have becomes 2x equals to log 9 * 8 over log 9.

[00:02:28]And this follows the law of logarithm.

[00:02:30]When we have log a base b, it's same thing as log a plus log b.

[00:02:42]And this here, we have 2x equals to log 9 plus log 8 over log 9.

[00:03:00]Okay. This follows when we have a plus b over c, which is same thing as a over c plus b over c.

[00:03:09]Then here, this becomes 2x equals to log 9 over log 9 plus log 8 over log 9.

[00:03:27]And this log 9 cancels each other here. We have one left, which becomes 2x equals to 1 plus log 8 over log 9.

[00:03:45]Then [snorts] from here, we can write 8 as 2 raised to power 3.

[00:03:51]And also nine raise three raise to power two.

[00:03:56]This equation becomes 2x equals to 1 + log two raise to power three over log three raise to power two.

[00:04:10]Apply the power rule of logarithm here.

[00:04:14]Three comes here, two comes here. This give us 2x equals to 1 + three log two over two log three.

[00:04:33]Which also can be written as 2x equals to 1 + three over two times log two over log three.

[00:04:46]And when we apply change of base, this when we have log A over log B, this is same thing as log A to base B.

[00:05:00]Then what we have here becomes 2x equals to 1 + three over two log two base three.

[00:05:12]I can bring everything to one side to one fraction. This one over one. We have 2x here equals to the LCM and that's two.

[00:05:21]This becomes two plus three log two base three.

[00:05:28]Then multiply both sides by one over two.

[00:05:31]Multiply this by one over two.

[00:05:34]So, multiply this by one over two.

[00:05:36]Here, two cancels each other here. We have X equals to two plus three log two base three over two times two here, that's four.

[00:05:50]So, we have the value of x in this problem in terms of logarithm. Then, let's check if this satisfy this given problem.

[00:06:06]That is, we substitute the value of x here, which is x equals to 2 + 3 log 2 base 3 over 4.

[00:06:25]Then, this equation already can write as 3 * 9 raised to power x.

[00:06:32]Sorry.

[00:06:34]That's 9 raised to power 2x rather.

[00:06:38]It's multiplication here.

[00:06:41]This equals to 72.

[00:06:43]So, when we substitute x here, this becomes 9 raised to power 2 * 2 + 3 log 2 base 3 over 4.

[00:06:59]Is it equals to 72 on this side?

[00:07:04]Then, yeah, since this power multiplies, 2 goes there 1, 2 goes there 2.

[00:07:09]And we have 9 raised to power 2 + 3 log 2 base 3 all over 2.

[00:07:18]Is it equals to 72 on this side?

[00:07:22]Also, 9 same thing as 3 squared.

[00:07:26]It is raised to power 2 + 3 log 2 base 3 all over 2. Is it equals to 72 on this side?

[00:07:37]And this power multiplies as well. 2 cancels each other here. We have 3 raised to power 2 plus 3 log 2 base 3 is it equals to 72 on this side.

[00:07:51]This follows law of indices. Can write as 3 squared times 3 raised to power 3 log 2 base 3 is it equals to 72 on this side.

[00:08:04]Then 3 squared is 9.

[00:08:07]Then we reverse this 3 whole. We have 3 raised to power log 2 raised to power 3, which is 8 base 3 is it equals to 72 on this side.

[00:08:19]Then this becomes 9 times this follows and we have a raised to power log b base a, which is equals to b.

[00:08:28]This here we have 8.

[00:08:30]Is it equals to 72 from here.

[00:08:34]Then 9 times 8 equals 72, which is equals to 72 here.

[00:08:38]Left hand side equals right hand side.

[00:08:43]And therefore we conclude that the value of x which is 2 plus 3 log 2 base 3 all over 4 satisfy this problem. Thank you for watching.

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