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

[00:00:00]In this video, let us solve for K given 5 raised to power K is equal to 500.

[00:00:21]We're given 5 raised to power K is equal to 500.

[00:00:30]Looking at 500, I'm going to break it down.

[00:00:38]500 divided by 5 here will give me 100.

[00:00:44]100 divided by 5 here will give me 20.

[00:00:48]And we know that 20 is 5 * 4.

[00:00:54]So, this is 5 * 5 * 5, which is 125 * 4.

[00:01:01]So, we can rewrite our equation as 5 raised to power K is equal to 125 * 4.

[00:01:12]I'm going to divide both sides by 125.

[00:01:21]So, that this here will take care of this, giving us 5 raised to power K divided by 125 is equal to 4.

[00:01:35]This is same thing as saying 5 raised to power K divided by 125 is 5 raised to power 3 then equal to 4.

[00:01:47]Let us apply this property of exponent given A raised to power M divided by A raised to power N by law of indices, this will give us a raised to the power m minus n.

[00:02:04]So, that this becomes 5 raised to the power k minus 3 is equal to 4.

[00:02:21]This is an exponential equation. Let us take the logarithm of both sides.

[00:02:26]So, log 5 raised to the power k minus 3 is equal to log 4.

[00:02:36]This expression here is of the form log p raised to the power a.

[00:02:42]By law of logarithm, this will give us a times log p.

[00:02:48]So, this becomes k minus 3 times log 5 is equal to log 4.

[00:03:00]Then, I'm going to divide both sides by log 5.

[00:03:09]So, that log 5 cancels log 5 here leaving us with k minus 3 is equal to log 4 divided by log 5.

[00:03:24]So, we have k minus 3 is equal to log 4 itself is 2 raised to power 2 then divided by log 5.

[00:03:38]Then, we have k minus 3 is equal to 2 times log 2 divided by log 5.

[00:03:52]Log 2 divided by log is of the form log a / log b.

[00:04:01]This will give us log a base b. So, this becomes log 2 base 5.

[00:04:07]And then we have k - 3 is equals to 2 * log 2 base 5.

[00:04:18]Moving this negative 3 to the other side will give us k is equal to positive 3 + 2 log 2 base 5.

[00:04:30]Which will now be our final answer to this problem.

[00:04:34]Let us now do a quick check to confirm that this is correct.

[00:04:39]To check, I'm going to substitute this value for k in our given equation, 5 raised to power k is equal to 500.

[00:04:50]This will imply 5 raised to power k is 3 + 2 log 2 base 5 to give us 500.

[00:05:10]Before we proceed, we need to separate these powers. And to do that, I use this law of indices.

[00:05:15]a raised to power m + n This will give us a raised to power m * a raised to power n.

[00:05:24]So, this expression becomes 5 raised to power 3 * 5 raised to power 2 log 2 base 5 to give us 500.

[00:05:39]5 raised to power 3 is 125.

[00:05:42]Then, if we bring this here by law of logarithm we get 5 raised to power log 2 raised to power 2 base 5 to give us 500.

[00:05:58]This gives us 125 times 5 raised to power log 2 raised to power 2 is 4 and base 5 to give us 500.

[00:06:13]Now, looking at this expression here, this is of the form a raised to power log m base a.

[00:06:21]By log logarithm, this will give us m.

[00:06:25]So, this is going to be 4.

[00:06:28]Then we have 125 times 4 to give us 500.

[00:06:39]125 * 4 is 500.

[00:06:42]So, 500 is equal to 500.

[00:06:46]And since the left-hand side balances the right-hand side that will confirm that the value we got for k which is this is perfectly correct.

[00:06:58]Thanks for watching. Please like and share and also remember to subscribe to my channel and I'll see you in my next video.

[00:07:06]Bye.