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

追加: 2026-05-30

[00:00:01]In this video, let us solve for K given 5^ K + 5^ K is equal to 100.

[00:00:22]We are given 5 raised to power k + 5 raised to power k is = 100 5 to power k is common here. We can factoriize it out as 5^ k then into brackets one and then we still have one here as well is equal to 100.

[00:00:51]This will give us 5 raised to power k * 2 is = 100.

[00:01:01]We can divide through by two on both sides to get rid of this two.

[00:01:08]Two here is one. 2 in 100 is 50.

[00:01:13]So we have 5 raised to power k is equal to 15.

[00:01:23]Now this is an exponential equation. So we can take the logarithm of both sides to give us log 5 to power k is equal to log 50.

[00:01:38]The left hand side expression is of the form log P raised to power C and by logarithm this will give us C * log P.

[00:01:52]So our equation becomes K * log 5 = log 50.

[00:02:05]Let us divide both sides by log five.

[00:02:15]This takes care of this leaving us with K is equal to log 50 / log 5.

[00:02:30]We can also write this as K is equal to log 50 here will give us 25 * 2 then divided by log 5.

[00:02:52]This expression here is of the form log a * b and by log logarithm this will give us log a plus log b.

[00:03:08]Therefore k is equal to log 25 + log 2 then divided by log 5.

[00:03:26]We can now separate this division to give us k= to log 25 / log 5 then plus log 2 / log 5.

[00:03:52]So k is equal to let us express 25 as 5 raised to power 2 then / log 5 then plus log 2 / log 5.

[00:04:12]So that k is now 2 * log 5 / log 5 + log 2 / log 5.

[00:04:35]Looking at this expression here, we see that log five divides log five here. So we have k is equal to we are left with two here then plus log 2 / log 5.

[00:04:56]log 2 / log 5 is of the form log a / log b.

[00:05:05]This will give us log a base b.

[00:05:10]And then we can now express k as 2 plus this will now be log 2 b 5.

[00:05:22]And all of this will now be our final answer to this problem.

[00:05:28]Let us now do a quick check to confirm that this solution is correct.

[00:05:34]To check we substitute this value of k back into the given problem which was 5^ k + 5^ k equal to 100 I can substitute individually but then this gave us 5^ k * 2 earlier = to 100. So it's now easier to substitute in one place this value of k.

[00:06:05]This will then imply 5 raised to power k is 2 + log 2 base 5 then * 2 to give us 100.

[00:06:26]Let's separate these powers. To do that we apply this law of indices P raised to power a + n.

[00:06:35]By law of indices this will give us p^ a * p ra.

[00:06:43]So this here gives us 5 to power 2 * 5 raised to power log 2 base 5 then* 2 is equal to 100 5^ 2 is 25 then I'll bring this forward here then times 5 raised to power log 2 is 5 to give us 100.

[00:07:18]25 * 2 is 50.

[00:07:22]Then times all of this 5 to power log 2 is 5 is equal to 100.

[00:07:31]5 raised to power log 2 5 here is of the form p raised to power log m p by law of logarithm this will just give us m so similar to this this will just give us two then we have 50 * 2 here to give us 100.

[00:08:01]50 * 2 is 100.

[00:08:05]So we have 100 is equal to 100. And since the left hand side balances the right hand side, that confirms that this value we got for K is absolutely correct. Thanks for watching. Please like and share and also remember to subscribe to my channel and I'll see you in my next video. Bye.