Proof of Log Properties: Rule 1

00:01 So let's now prove the first rule of logs.

00:04 We said earlier that log a of xy equals to log base a of x plus log base a of y.

00:14 I've wrote with the other way around but it'll be easier for us to prove it this way.

00:18 Now, we can start off by changing these to some easier letters.

00:23 So let's just say, let p equals to log base a of x, so that's that term there and let q equals to log base a of y.

00:35 Now one of the things that you can do with logs is change it to index form.

00:41 So if we have something like log base a of x equals to y, you can rewrite this as indices so as an index form.

00:51 So this a value becomes your base, this becomes your power and this is your answer.

00:58 If you've done very little of logs before, you may have already seen this but this could be another one of those rules that you can just add into your collection of rules that if you have log base a of x equals to y, you can write this into index form.

01:15 So we usually call this log form, so this is in log form and this here is index form and it's a nice little transaction that you can make between the two.

01:26 You can change logs to exponential or index form and index to log form.

01:31 So using that rule, I can rewrite these terms here so I can re-write these as a to the power of p equals to x, so remember this is my base number, this is my power and this is my answer just based on what we've just said here and you can do the same here base a to the power of q equals to y.

01:52 So essentially, we're saying that x is a to the p and y is a to the q.

01:58 Right, let's came back to our statement here, this is what we want to prove, you can start to prove it one a little bit at a time so let's just start with xy.

02:08 So we can say, so x multiplied by y. So this x and this y, so x is a to the p and y is a to the q.

02:18 When two numbers or the two base number that the same, you can add the power so that becomes a, p plus q.

02:25 So far we're saying that xy equals to a to the p plus q.

02:31 You'll notice here that we've got the xy but you also have a log on the outside, so let's just log both sides.

02:39 So if I do it to one side, let's just do it to the other side.

02:42 So if I have log base a, wanna make some room here.

02:45 So I have a to the power of p plus q and I also want to log base a here.

02:52 So if I do it to the same side, I'm not essentially changing the equation.

02:57 Okay, from here now, you can find that if you bring the p plus q to the front using the third rule of log, we're now left with log a here and then you have log base a of xy, so remember I just used the third rule of log here, even just brought it down to the front.

03:17 Another little thing or another little rule of logs is that if you have log base a of a that equals to 1.

03:27 Now this is just something you can try out on your calculator, see if you can try log base 10 of 10, you'll get an answer of 1.

03:34 If you did log base e which is basically ln of e, that will give you answer of 1 as well.

03:42 So this is just something that doesn't really need a proof so it's something that you can learn about logs that if you have log base number on the number next to it being the same, it's just equals to 1.

03:53 So if this equal to 1, we now have log base a, xy equals to p plus q and remember what we said right at the start we said p is this value and q is this value.

04:06 So if we just put that back in into here, log base a of x plus log base a of y, we have proved that log base a of xy is log base ax plus log base a of y.