Only 1% Got This Viral Math Olympiad Challenge! | Can You Solve It?

[00:00:01]Hello, you're welcome. I do solve this nice exponential equation to find the value of n here.

[00:00:08]Solution from here.

[00:00:12]And what we have which is 8 raised to the power n equals to 120.

[00:00:19]First step here, we take the log on both sides. That is, we have log 8 raised to the power n equals to log 120 on this side.

[00:00:34]And this follows the power rule of logarithm. When we have log m raised to the power p, we write it as p log m.

[00:00:45]And here, the power is n. And this we can write it as n log 8.

[00:00:52]Equals to log 120 on this side.

[00:00:59]The next step, divide both sides by log 8. That is, divide this side by log 8.

[00:01:05]Also, divide this side by log 8.

[00:01:10]Which implies that log 8 cancels each other.

[00:01:14]And we have n equals to log 120 over log 8.

[00:01:27]So, and we can express 120 as a product of its prime. Now, we divide by the first prime number, which is 2.

[00:01:37]So, we have 2 goes in 120, that's 60.

[00:01:42]2 goes in 60 again, that's 30.

[00:01:46]So, 2 goes in 30 again, that's 15.

[00:01:50]And here, 3 will go next, that's 5. Then 5, that's 1.

[00:01:55]So, it means that here, 120 can be expressed as two times two times two times three times five.

[00:02:07]Which is also same thing as two times two times two, that's two raised to power three times three times five.

[00:02:16]Which implies this equation becomes n equals to log two raised to power three times three times five over log eight.

[00:02:34]So, this here follows the law of logarithm. When we have log a times b times c I write this as log a plus log b plus log c.

[00:02:54]Which implies we have n equals to log two raised to power three plus log three plus log five all over log eight.

[00:03:15]Then this follows when we have log Okay? When we have a plus b plus c over d, I can write this as a over d plus b over d plus c over d.

[00:03:34]So, this here also we have n equals to log two raised to power three over log eight plus log three over log eight plus log 5 over log 8.

[00:03:58]Then, next step here also, we can express it as 2 raised to power 3.

[00:04:06]Which means this equation becomes n equals to log 2 raised to power 3 over log 2 raised to power 3 plus also log 3 over log 2 raised to power 3 plus log 5 over log 2 raised to power 3.

[00:04:32]So, yeah, it's obvious that these are the same things, so these can cancel each other.

[00:04:38]You have one here.

[00:04:40]This n equals to 1 plus Applying the power rule of logarithm, 3 comes here and also here. So, we have log 3 over 3 log 2 and plus log 5 over 3 log 2.

[00:05:01]So, this is also the same thing as n equals to 1 plus 1 over 3 times log 3 over log 2 plus also 1 over 3 times log 5 over log 2.

[00:05:23]Then, applying change of base here, we have log a over log b.

[00:05:30]This is the same thing as log a base b.

[00:05:35]So, what we have here becomes n equals to 1 plus 1 over 3 log 3 base 2 plus 1 over 3 log 5 base 2.

[00:05:52]I'm going to bring this together as one fraction. This gives us n equals to this one over one and the LCM here is three.

[00:06:01]So, here we have three and plus log 3 base 2 plus log 5 base 2. So, we have the value of n in terms of logarithm here in this problem.

[00:06:17]And we can check if this satisfies this problem. We substitute the value of n here which is n equals to 3 plus log 3 base 2 plus log 5 base 2 all over 3.

[00:06:43]Then this equation becomes 8 raised to power 3 plus log 3 base 2 plus log 5 base 2 all over 3.

[00:06:59]This is equals to 120 on this side.

[00:07:03]Then, from here we can write three eight here rather as two raised to power three which is raised to power three plus log 3 base 2 plus log 5 base 2 all over 3.

[00:07:22]This is equals to 120 on this side.

[00:07:27]Then here, this power multiplies. Three cancels each other.

[00:07:31]And we have two raised to power three plus log 3 base 2 plus log 5 base 2.

[00:07:42]This is equals to 120 on this side.

[00:07:47]Now we apply the law of indices. And so it brings us 2 raised to power 3 times 2 raised to power log 3 base 2 times 2 raised to power log 5 base 2 This is equals to 120 on this side.

[00:08:06]Then 2 raised to power 3, that's 8. Then this one also we have a raised to power log b to base a is same thing as b.

[00:08:19]Now write this as 3 and this this one as 5 is equals to 120 here.

[00:08:26]So yeah, 8 times 3, that's 24. And 24 times 5, that's 120.

[00:08:34]It is equals to 120 here.

[00:08:37]Which implies left hand side equals to right hand side.

[00:08:42]And therefore it concludes that the value of n here it is 3 plus log 3 base 2 plus log 5 base 2 all over 3 satisfy this given problem. Thank you for watching. Don't forget to subscribe for more videos and turn on the notification bell.

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