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

Added: 2026-05-25

[00:00:01]In this video, let us solve for a given 4^ a * 16 power a= 40.

[00:00:24]We're given 4 raised to power a * 16 raised to power a is = 40.

[00:00:38]Looking at 16 here we can express it as 4 raised to power 2. So we have 4 to power a first then 4 to power 2 then raised to power a is equal to 14.

[00:00:59]We can apply law of indices to this expression. Given a raised to power p then raised to power n by law of indices we can open this bracket to give us a raised to power p * n or n * p to give us np.

[00:01:21]So our equation becomes 4 raised to power a * 4 raised to power 2 * a 2 a is equal to 14.

[00:01:39]We need to apply another law of indices.

[00:01:42]Given a raised to power p time same a raised to power m by law of indices we can bring these two together to give us a raised to power p + m.

[00:01:58]Then applying this to the left hand side expression, this will give us 4 raised to power a + 2 a is = 40.

[00:02:13]Then we get 4 raised to power a + 2 a here is 3 a = 40.

[00:02:24]Now this is an exponential equation and we can take the logarithm of both sides.

[00:02:30]So we have log 4 raised to power 3 a is equal to log 40.

[00:02:42]log 4^ 3 a is of the form log a raised to power m which by law of logarithm will give us m * log a.

[00:02:58]So this becomes 3 a * log 4 is equal to log 40.

[00:03:11]Let us eliminate logarithm from the left hand side by dividing both sides by log four.

[00:03:24]So this here takes care of this leaving us with 3 a is equal to log 40 / log 4.

[00:03:43]We can also write this as 3 a is equal to log 40 here can also be written as 8 * 5 then divided by log 4.

[00:04:04]log 8 * 5 is of the form log m * p which by love logarithm will give us log m + log p.

[00:04:22]Then our equation becomes 3 a is = log 8 + log 5 then divided by log 4.

[00:04:46]We can separate this division into log 8 / log 4 + log 5 / log 4.

[00:05:11]And then this becomes 3. A is equal to let us express 8 in index form. This will give us 2^ 3. We do the same thing for this four here. So that this gives us log 2 raised to power 2 then plus log 5 / log 2 raised to power 2.

[00:05:47]This then becomes 3 a is equal to we can write this by love logarithm as 3 log 2 then divided by this also we write as 2 log 2 then plus log 5 / this is 2 log 2 Then we see that log 2 cancels log 2 here leaving us with 3 a is equal to 3 / 2 plus we can rewrite this as 1 / 2 * log 5 / log 2.

[00:06:40]Looking at log 5 / log 2. This is of the form log p / log m and by law of logarithm this will give us log b m.

[00:06:57]So we have 3 a is = 3 / 2 + 1 / 2. This can now be expressed as log 5 b 2.

[00:07:14]So we have log 5 b 2.

[00:07:18]And the final thing we need to do to this equation will be to divide both sides by three.

[00:07:24]That is all of this / 3. 3 takes care of three here. So that a will now be equal to 3 / 2 / 3 that is 3 / 6 then + 1 / 6 log 5 base 2 and finally three here is 1 3 here is 2.

[00:07:53]So we have a is = 1 / 2 + 1 / 6 log 5 base 2.

[00:08:07]This gives us the final answer to this problem.

[00:08:13]We will now do a very quick check to confirm that this is correct.

[00:08:21]So we need to substitute this value of a back into the given problem. 4 to power a * 4 16 raised to power a is = 40.

[00:08:38]But before we go ahead and make this substitution, we simplify this into 4^ 3 a then equal to 40.

[00:08:49]So we can use this for our check to make the check process a bit simpler.

[00:08:56]So we have a is all of this and we are substituting that into this equation. If the left hand side is equal to the right hand side, it confirms that this solution is correct.

[00:09:10]This would then imply 4 raised to power 3 into bracket is 1 / 2 + 1 / 6 log 5 is 2 to give us 40.

[00:09:36]Let us open up this bracket. This will give us 4 raised to power 3 * 1 / 2 that is 3 / 2 + 3 * 1 / 6 that's 3 / 6 which will be 1 / 2.

[00:09:51]Then time log 5 base 2 is equal to 40.

[00:09:59]Next thing we'll do will be to apply law of indices to separate these powers.

[00:10:05]Given P raised to power m + n by law of indices this will give us p raised to power m * p raised to power n.

[00:10:17]Therefore this becomes 4 raised ^ 3 / 2 * 4 raised to power 1 / 2 log 5 this 2 to give us 40.

[00:10:36]This is same thing as saying<unk> 4 raised to power 3 time<unk> 4 raised to power log 5 base 2 to give us 14 square root of 4 is 2. So this will be 2 raised ^ 3 * 2 raised to power log 52 to give us 40 2^ 3 is 8 times all of this 2 power log 5 is 2 to give us 14.

[00:11:27]Notice that we have two here. We also have two here. By law of logarithm given p raised to power log m p. As long as these are the same, we have p here and p here. This will always simplify to m.

[00:11:49]Therefore, our expression here will give us five.

[00:11:54]Then we get 8 * 5 to give us 40.

[00:12:03]8 * 5 is 40. So we have 40 here is equal to 40 here. And therefore the left hand side balances the right hand side which confirms that this value that we got for a is absolutely correct.

[00:12:22]Thanks for watching. If you have enjoyed this video, please like and share and also remember to subscribe to my channel and I'll see you in my next video. Bye.