Hi guys and welcome now, in order for you to truly take charge of logs, you need to
understand this really cool tool, it's called the 'Change of Base Rule'.
Now what does that do for us?
It simply allows us to convert or transform a logarithmic expression from one format into
another, in other words we can easily change the bases.
So if we start off with a base of 7, we can change it into a base of 10 if we want which
is very convenient because most calculators have base 10, so if you don't have the base
on your calculator or it's an unfamiliar base, this is a really cool tool that let's you
change into any base that you want, so let's do the proof so we can understand this rule
a lot more deeper.
So how do we start with this proof, how do we get to this place right over here?
Well first of all we start off with a fundamental logarithmic expression, now many of you have
already seen this on the other videos, it's X= log base A, base being the basement being
the base below and the number.
Now 'X' is usually the power or the exponent over there on the left.
We're going to use this and we're going to change it into index or exponent form and
start to do an expression down here and hopefully we're going to come up with this rule right
over there, so let's see how we go now.
So the first thing we want to do is take this and transform it into what format?
Exponent or Power Format, so let's do that now.
So the first thing I'm going to do is write it up as follows, I'm going to write up over
here 'N' that's the number.
So 'N' the number= A the base, now the base suddenly becomes bigger, so it's A the base
and that's going to be to the power of X.
so all we simply do is as many of you probably already know, that base goes under this 'X'
power and we get this kind of a format.
So there it is in Power Format, A (number) = A base to the power of X.
So what we're going to do now is we're going to log both sides, we're going to take a log
of both sides.
Being an equation takes balance so what you don on this side, you do it on that side.
So let's take a log on both sides and what log would we like to do?
Well let's introduce a new log, a new base sorry.
If you've got base A, let's change it into say base 'B', so let's do this, log base 'B',
no log base 'B' on the other side, there we go, we've just introduced a new base, base
'B', base 'B', and that's where we going.
So what can we do for the next step?
Well we can simplify the right hand side by using the Power Law, if you remember the log
laws, the Power Law where we can take the Exponent of the power and bring it to the
front over here and it still remains the same and that's the power right over here.
So let's do that now... on the left hand side, it's still log base 'B', then number 'N' and
that equals 'X' log... there's the new base right there, the two little 'Bs', so it would've
changed from 'A' to 'B'.
So let's further simplify this.
All you have to do now is make this 'X' the subject, so let's make it the subject and
bring it across to the left, let's kind of like swap them around, so let's bring the
'X' over here, so 'X' is going to equal to what, we're going to divide, we're going to
drag and drop the log base 'B' 'A' underneath log base 'B' 'N', let's do that, so let's
write this top part over here, so on the top we have log base 'B' 'N', that's going to
be all over right here, log base 'B' and that's going to be base 'B' 'A' right there.
So as you can see, we have the Exponent or the Power, equaling log base 'B' 'N' which
what our outcome is to have a base of 'B' a new base over log base 'B' 'A', now that's
almost exactly the same as that rule right over there and that's our proof and stay tuned
we're going to do some examples, so we can get deeper clarity in this.
Thanks for watching.