Solve for x in this nice Algebra equation | Math Olympiad Mathematics

追加: 2026-05-28

[00:00:01]In this video, we want to solve for X.

[00:00:07]Given 5 raised to the power X divided by 25 is equal to 50.

[00:00:27]We're given 5 raised to the power X divided by 25 is equal to 50.

[00:00:38]First thing I'll do here will be to cross-multiply so that I have 5 raised to the power X times 1 is equal to 25 here times 50.

[00:00:59]This will give us 5 raised to the power X is equal to 25 here is 5 raised to the power 2.

[00:01:10]Then 50 is same thing as saying 25 times 2.

[00:01:19]So we have 5 raised to the power X is equal to 5 raised to the power 2 times 5 raised to the power 2 times 2.

[00:01:32]This is same thing as saying 5 raised to the power X equal to 5 raised to the power 2 raised to the power 2 then times 2.

[00:01:45]Given P raised to the power A raised to the power M by law of indices, this will give us P raised to the power A times M.

[00:01:56]So we can rewrite this equation as 5 raised to power x is equal to 5 raised to power 2 * 2, which will be 4, then * 2.

[00:02:09]Let us divide both sides by 5 raised to power 4.

[00:02:18]This here takes care of this, leaving us with 5 raised to power x divided by 5 raised to power 4 is equal to 2. Let us apply another law of indices to this left-hand side expression.

[00:02:36]This expression is of the form p raised to power a divided by p raised to power m.

[00:02:44]By law of indices, this will give us p raised to power a minus m.

[00:02:50]Therefore, we have 5 raised to power x minus 4 is equal to 2.

[00:03:02]What we have now is an exponential equation.

[00:03:05]To proceed from here, let us take the logarithm of both sides.

[00:03:11]So, we have log 5 raised to power x minus 4 is equal to log 2.

[00:03:21]This here is of the form log p raised to power c and by law of logarithm, this will give us c times log p.

[00:03:35]So, we have x minus 4 times log 5 is equal to log 2.

[00:03:49]Let us divide both sides by log 5.

[00:03:59]This here takes care of this.

[00:04:03]Leaving us with x - 4 is equal to log 2 divided by log 5.

[00:04:17]Log 2 divided by log 5 is of the form log a divided by log b.

[00:04:26]And by law of logarithm, this will give us log a base b.

[00:04:32]Therefore, we now have x - 4 is equal to log 2 base 5.

[00:04:44]I'll transfer -4 to the other side.

[00:04:47]It will become positive. So, we have positive 4 then plus log 2 base 5.

[00:04:57]This will then be the final answer to this problem.

[00:05:02]Before we conclude, let us do a quick check to confirm that the solution is correct.

[00:05:08]And to do that, we'll simply substitute this value of x back into the given problem, which was 5 raised to power x divided by 25 is equal to 50.

[00:05:25]This will imply 5 raised to power x is 4 plus log 2 base 5 then divided by 25 to give us 50.

[00:05:47]Let us separate these powers by law of indices given P raised to power A plus C.

[00:05:54]This will give us P raised to power A times P raised to power C.

[00:06:00]So, this becomes 5 raised to power 4 times 5 raised to power log 2 base 5.

[00:06:13]All of that divided by 25 to give us 50.

[00:06:23]Then this is 5 raised to power 4.

[00:06:28]By law of logarithm given P raised to power log M base P this will always give us M.

[00:06:44]Therefore this right here will give us 2.

[00:06:49]So, that we now have 5 raised to power 4 times 2 divided by 25 is equal to 50.

[00:07:06]So, 5 raised to power 4 is going to give us 5 times 5 times 5 times 5 then times 2 divided by 25 to give us 50.

[00:07:24]So, 5 times 5 is 25. That takes care of this. So, we have 5 times 5 times 2 to give us 50.

[00:07:38]This will give us 50.

[00:07:42]There we have 50 is equal to 50.

[00:07:45]Therefore, the left-hand side balances the right-hand side. And that confirms that this solution we got for X here is absolutely correct.

[00:07:55]Thanks for watching. Please like and share, and also remember to subscribe to my channel if you have not done so already. And I will see you in my next video.

[00:08:04]Bye.