Why Everyone Is Failing This Simple Olympiad Question

Added: 2026-05-11

[00:00:00]how to solve this math Olympiad question.

[00:00:03]This is very simple to solve and I want you guys to pay close attention.

[00:00:08]Because by the end of this video, you will know the solution to this equation.

[00:00:12]So, let's get started.

[00:00:14]Now, from the law of indices in the left-hand side, if you have a to the power of x plus y, this can be split and it can be written as a to the power of x multiplied by a to the power of y.

[00:00:35]So, in application of this in this expression, this become 5 to the power of x multiplied by 5 to the power of 1 to be equal to 6 to the power of x.

[00:00:50]So, 5 to the power of 1 is 5. So, we shall have 5 to the power of x multiplied by 5 to be equal to 6 to the power of x.

[00:01:01]Now, there's something I want to do to this equation. Very interesting. Watch.

[00:01:07]I want to shift this unknown to one side.

[00:01:10]So, how can I achieve that? I can achieve that by dividing both side by 6 to the power of x.

[00:01:16]So, if I divide here by 6 to the power of x, also divide here by 6 to the power of x.

[00:01:23]Okay? So, I want to separate this this left-hand side.

[00:01:29]This can be written as 5 to the power of x divided by 6 to the power of x multiplied by 5 to be equal to 1.

[00:01:39]Because this divide this is 1.

[00:01:43]So, what next?

[00:01:46]Um there's a law of indices that state that if you have um a to the power of x divided by b to the power of x, this is the same thing as saying A over B all to the power of X.

[00:02:02]So, in the application of this law in this place, we shall have five over six all to the power of X multiplied by five is equal to one.

[00:02:15]Now, watch this.

[00:02:17]I don't want this five to be in the left-hand side. So, what can we do?

[00:02:22]Let's multiply or let's divide both side by five.

[00:02:26]So, if we have five over six all to the power of X multiplied by five is equal to one.

[00:02:35]If we divide here by five and also divide here by five.

[00:02:38]Okay?

[00:02:39]So, this will cancel out. We're going to leave it five over six all to the power of X will be equal to one over five.

[00:02:50]Now, we cannot express the base to be in the same form.

[00:02:53]What do you do? You introduce log to both side. That's logarithm to both side.

[00:02:59]So, we shall take the log of both side in base of five over six.

[00:03:04]So, we shall have the log of five over six all to the power of X in base of five over six will be equal to the log of one over five in base of five over six. As you can see, I am introducing log of five base six to both side.

[00:03:25]So, what we are going to do is that there's a law of logarithm that state that if you have um the log of A to the power of B multiplied by A, this is the same thing as saying B log of A base of A. Of course, the log of A base of A is equal to one. So, we shall have B times one which is equal to B.

[00:03:50]So, X will multiply the whole this, which become X multiplied by the log of 5 over 6 base of 5 over 6 will be equal to the log of 1 over 5 base of 5 over 6.

[00:04:07]So, the whole of this become 1 according to this. So, we shall have X times 1 is X. So, which become X will be equal to the log of 1 over 5 base of 5 over 6.

[00:04:20]And this happens to be the final answer.

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