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

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

[00:00:08]Solution from here.

[00:00:11]What we have which is 4 raised to power X over 10 equals to 10.

[00:00:20]First step here, multiply both sides by 10. That is, multiply this side by 10.

[00:00:25]Also, multiply this side by 10.

[00:00:29]Here, 10 cancels to each other here. We have 4 raised to power X equals to 10 * 10 here. I write it as 100.

[00:00:38]Then here, take the log on both sides.

[00:00:41]This gives us log 4 raised to power X equals to log 100 here.

[00:00:52]Then this follows the power law of logarithm. When we have log M raised to power P is the same thing as P log M.

[00:01:04]Then also, what we have here becomes X log 4 equals to log 100 here.

[00:01:15]The next step, divide both sides by log 4. I just divide this side by log 4.

[00:01:22]Also, divide this side by log 4.

[00:01:25]Which implies log 4 canceled each other here.

[00:01:30]We have X equals to log 100 over log 4.

[00:01:41]The next step here, 100 can be expressed as 4 * 25.

[00:01:50]Which implies what we have becomes X equals to log 4 * 25.

[00:02:00]over log 4.

[00:02:06]The next step here, this follows a law of logarithm where we have log a * b.

[00:02:13]We can write this as log a plus log b.

[00:02:20]And this here, we have x equals to log 4 plus log 25 over log 4.

[00:02:35]And this follows how we have a plus b over c.

[00:02:40]This engine as a over c plus b over c.

[00:02:45]And this also can raise as x equals to log 4 over log 4 plus log 25 over log 4.

[00:03:03]Which implies from here, log 4 cancels with each other. We have one left.

[00:03:08]This becomes x equals to 1 plus log 25 over log 4.

[00:03:20]The next step here we can write 25 as 5 * 5.

[00:03:27]It is the same thing as 5 squared.

[00:03:30]And also 4 as 2 * 2.

[00:03:34]It is the same thing as 2 squared.

[00:03:37]So, this equation becomes x equals to 1 plus log 5 squared over log 2 squared.

[00:03:51]Then we apply the power rule of to 2 ^ x and also here, this becomes x equals to 1 plus 2 log 5 over 2 log 2.

[00:04:06]Now this here 2 cancels each other and we have x equals to 1 plus log 5 over log 2.

[00:04:21]So also we can rewrite this And this we apply change of base here, and we have log a over log b is the same thing as log a to base b. So here we have x equals to 1 plus log 5 base 2. So we have the value of x here in terms of logarithm.

[00:04:50]That's 1 plus log 5 base 2. Let's check if this satisfies this given problem.

[00:04:56]We substitute the value of x here which is x equals to 1 plus log 5 base 2.

[00:05:15]That is this equation becomes 4 raised to the power 1 plus log 5 base 2 over 10.

[00:05:26]This is equals to 10 on this side.

[00:05:31]And here we can express 4 as 2 squared which is raised to the power 1 plus log 5 base 2 and over 10 is equals to 10 on this side.

[00:05:52]Then here, 2 upon this bracket, we have 2 raised to power 2 plus 2 log 5 base 2 all over 10.

[00:06:04]This is equals to 10 on this side.

[00:06:09]Then applying the law of indices here, this a raised to power a plus n is the same thing as a raised to power a times a raised to power n.

[00:06:19]But this a we have 2 squared times 2 raised to power 2 log 5 base 2 over 10.

[00:06:29]This is equals to 10 on this side.

[00:06:33]As this here 2 squared that's 4 times we divide this 2 over here 2 raised to power log 5 squared which is 25 base 2 over 10.

[00:06:46]This is equals to 10 on this side. So here, this follows the law of logarithm.

[00:06:53]a raised to power log b to base a is the same thing as b.

[00:06:59]Now we compare this here. We can write this as 4 times 25 over 10 is is equals to 10 on this side.

[00:07:11]Which implies here, 5 goes in 10 2. 5 goes in 25 5.

[00:07:17]2 goes in 2 1. 2 goes in 4 2.

[00:07:21]And we are left with 2 times 2 which is 10.

[00:07:25]10 equals to 10 on this side.

[00:07:27]Left hand side equals to the right hand side.

[00:07:32]That is we can conclude that the value of x here which is 1 plus log 5 is 3 satisfy this given equation.

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