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

Added: 2026-05-20

[00:00:00]In this video, let us solve for M given 3 raised to power M * 3 raised to power M is equal to 48.

[00:00:22]We are given 3 raised to power M * 3 raised to power M is equal to 48.

[00:00:34]By law of indices, we can bring these two together to give us 3 raised to power M raised to power 2 is equal to 48.

[00:00:48]This expression here is of the form P raised to power X raised to power A.

[00:00:57]We can expand this using law of indices this to give us P raised to power X * A here will give us AX.

[00:01:07]So, this becomes 3 raised to power M * 2 2M is equal to 48.

[00:01:19]This is an exponential equation.

[00:01:21]Let us take the logarithm of both sides.

[00:01:25]So, we have log 3 raised to power 2M is equal to log 48.

[00:01:35]Log 3 raised to power 2M is of the form log A raised to power C.

[00:01:43]By law of logarithm, this will give us C * log A.

[00:01:49]Therefore, this will give us 2M * log 3 is equal to log 48.

[00:02:04]Let us get rid of log 3 from the left-hand side.

[00:02:07]We'll divide through by log 3 on both sides.

[00:02:14]Then, this here takes care of this, giving us 2m is equal to log 48 divided by log 3.

[00:02:31]We can rewrite this as 2m is equal to log 48 here is 3 * 16 then divided by log 3.

[00:02:52]Given log a * b, by law of logarithm, this will give us log a plus log b.

[00:03:05]Therefore, our equation becomes 2m is equal to log 3 plus log 16 then divided by log 3.

[00:03:26]We can separate this division to give us 2m is equal to log 3 divided by log 3 plus log 16 divided by log 3.

[00:03:48]Log 3 divided by log 3 here will give us 1.

[00:03:51]So, we have 2 m is equal to 1 plus log 16 divided by log 3.

[00:04:06]Then, 2 m is equal to 1 plus We can write this as log 16 is 2 raised to power 4.

[00:04:18]Then, divided by log 3.

[00:04:26]Then, 2 m is equal to 1 plus This becomes 4 * log 2 divided by log 3.

[00:04:42]Which we can also write as 2 m is equal to 1 plus 4 * log 2 divided by log 3.

[00:04:59]Log 2 divided by log 3 here is of the form log a divided by log b.

[00:05:10]By law of logarithm, this will give us log a base b.

[00:05:17]So, that's how our equation becomes 2 m is equal to 1 plus 4 * This will now be log 2 base 3.

[00:05:34]Let us divide both sides by 2.

[00:05:36]That would be each of this also divided by 2.

[00:05:41]This takes care of this.

[00:05:44]So, we have m is equal to 1 divided by 2 plus 2 here is 1, 2 here is 2.

[00:05:54]So, this should be 2 log 2 base 3.

[00:06:00]Giving us the final answer to this problem.

[00:06:05]Now, let us do a quick check to confirm that this is correct.

[00:06:11]To check, let us substitute this value for m back into the original equation 3 raised to power m * 3 raised to power m is equal to 48.

[00:06:29]We simplify this to 3 raised to power 2 m is equal to 48.

[00:06:38]Now, in place of m here I'm going to put this which would then apply 3 raised to power 2 * 1 over 2 plus 2 log 2 base 3 to give us 48.

[00:07:13]If we open up this bracket this will give us 3 raised to power this takes care of this, so we have 1 plus 2 * 2 is 4 then times log 2 base 3 to give us 48.

[00:07:32]We need to separate these powers, so we use this law of indices p raised to power a plus b.

[00:07:39]By law of indices, this will give us p raised to the power of A.

[00:07:43]times P raised to the power of B.

[00:07:47]Which would then imply 3 raised to the power of 1 times 3 raised to the power of 4 times log 2 base 3 to give us 48.

[00:08:06]This is 3 times We carried down this all the way here by the law of logarithm.

[00:08:13]So that we have 3 raised to the power of log 2 raised to the power of 4 base 3 to give us 48.

[00:08:24]This would imply 3 times 3 raised to the power of log 2 raised to the power of 4 is 16.

[00:08:36]Then base 3 to give us 48.

[00:08:42]By the law of logarithm, this expression here which is of the form A raised to the power of log M base A to give us M.

[00:08:55]Therefore, this expression here will give us 16.

[00:09:00]Then we have 3 times 16 to give us 48.

[00:09:10]3 times 16 is 48.

[00:09:13]So we have 48 is equal to 48.

[00:09:18]Therefore, the left-hand side balances the right-hand side.

[00:09:21]And that confirms that the value we got for M is absolutely correct.

[00:09:27]Thanks for watching. Please like and share. And also remember to subscribe to my channel. And I'll see you in my next video.

[00:09:36]Bye.