Germany | Can You Solve This? | Math Olympiad

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[00:00:01]Hello, you're welcome. How do you solve this nice exponential equation?

[00:00:06]Solution from here.

[00:00:09]What is given? We factor 4 raised to power x out. We have 4 raised to power x into bracket.

[00:00:15]We are left with 1 + 1.

[00:00:18]Close bracket equals to 80 from here.

[00:00:23]Then, this becomes 4 raised to power x times 1 + 1 equals to equals to 80 on this side.

[00:00:33]Then, we divide both sides by two here.

[00:00:35]Divide this side by two. Also, divide this side by two. That is, two canceled each other here. Therefore, raised to power x equals to 80 divided by two.

[00:00:51]This from here, we can write this as 4 raised to power x equals to two times 40 over two.

[00:01:00]Here, two canceled each other.

[00:01:03]And that makes it 4 raised to power x equals to 40 from here.

[00:01:10]The next step, we take the log on both sides. That is, we have log 4 raised to power x equals to log 40 here.

[00:01:23]When we apply the power law of logarithm, log m raised to power p will give us p log m.

[00:01:33]That is, what we have becomes x log 4 equals to log 40 here.

[00:01:46]Next step, divide both sides by log 4.

[00:01:48]We divide this side by log 4. Then, also divide this side by log 4.

[00:01:54]That is, log 4 canceled each other here. And we have X equals to log 40 over log 4.

[00:02:10]Next step here, we can write 40 [snorts] as 4 * 10.

[00:02:16]Which implies we have X equals to log 4 times 10 over log 4.

[00:02:27]And this follows the law of logarithm where we have log A times B.

[00:02:32]I write this as log A plus log B.

[00:02:38]And this we have X equals to log 4 plus log 10 over log 4.

[00:02:52]Then we separate this into two fractions, and this becomes X equals to log 4 over log 4 plus log 10 over log 4.

[00:03:10]That is yeah, log 4 cancelled each other. We have one left, which implies X equals to 1 plus log 10 over log 4.

[00:03:25]And next step here, we can write 10 as 2 times 5.

[00:03:31]That is yeah, we have X equals to 1 plus log 2 times 5 over log 4.

[00:03:44]Then this also follows the law of logarithm, and this becomes X equals to 1 plus log two plus log five over log four.

[00:04:02]When we separate this into two fractions, we have x equals to one plus log two over log four plus log five over log four.

[00:04:23]Next step, yeah.

[00:04:24]Also, four can be expressed as two squared which implies we have x equals to one plus log two over log two squared plus log five over log two squared.

[00:04:46]When we apply the power of log in, two comes here and also here. Then we have x equals to one plus log two over two log two plus log five over two log two.

[00:05:08]That is from here, log two cancel with each other. This becomes x equals to one plus one over two plus Also here we have one over two times log five over log two.

[00:05:27]Then we can rewrite this as x equals to one plus one over two that's three over two plus one over two log five over log two.

[00:05:43]Then from here we apply change of base.

[00:05:46]When we have log a over log b, we can write this as log a to base b.

[00:05:57]That is, this becomes x equals to 3 over 2 plus 1 over 2 log 5 base 2.

[00:06:09]Here, we bring this together as one fraction, and we have x equals to the same denominator as 2.

[00:06:16]And we have 3 plus log 5 base 2. As this, we have the value of x in this problem in terms of log.

[00:06:29]Then, let's check if this satisfies this given problem. We substitute the value of x here, which is x equals to 3 plus log 5 base 2 all over 2.

[00:06:50]Then, this becomes 4 raised to power 3 plus log 5 base 2 all over 2 plus.

[00:07:00]Also, 4 raised to power 3 plus log 5 base 2 all over 2.

[00:07:08]Is it equals to 80 on this side?

[00:07:13]Then, when we had same things like a plus a, it's same thing as 2 times a.

[00:07:21]Also, we can write this as 2 times 4 raised to power 3 plus log 5 base 2 over 2.

[00:07:30]Is it equals to 80 on this side?

[00:07:34]Yeah, 4 can be written as 2 squared. We have 2 times 2 squared, which is raised to power 3 plus log 5 base 2 over 2 Is it equals to 80 on this side?

[00:07:50]This part multiplies 2 here cancel each other here.

[00:07:53]2 * 2 raised to power 3 plus log 5 base 2 Is it equals to 80 on this side?

[00:08:06]Then we can write this as 2 * 2 raised to power 3 then 2 raised to power log 5 base 2 Is it equals to 80 on this side?

[00:08:19]Okay, this becomes 2 * 2 raised to power 3 that's 8.

[00:08:24]Then this follows on we have a raised to power log b to base a this equals to b.

[00:08:31]But this a we have 5.

[00:08:34]Is it equals to 80 on this side?

[00:08:38]Then 8 * 5 40 40 * 2 it was 80 equals to 80 from here. Left hand side now equals to the right hand side and therefore we conclude that the value of x here which is 3 + log 5 base 2 all over 2 satisfy this given problem. Thank you for watching. Don't forget these steps.

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