Mr. Tarrou. We are going to talk for a minute
about the Change of Base or changing the base
of a logarithm. This formula log base b of
m equals log base a of m divided by the log
base a of b. So we have a logarithm originally
in base b form and now it is in log base a
form. For most of uf with normal calculators,
that normal base that we want is either natural
log, log base e, or log base ten the common
log. Why do we need this? I contend that you
don't even have to memorize this in my opinion
once you learn how to solve equations with
the variable in the exponent. But, it is a
standard formula and it is in all the precalculus
and trig books and so on. So, let's take a
look at a question like this. Log base four
of the cube root of four. Well, you don't
need a change of base formula for this even
though it is a base that is that is not a
common base, it is not base ten or base e
it is base four. But this one you can work
out with out the use of a calculator like
my students are going to be doing tomorrow.
Ok, what does this equal? Let's say that I
don't know. Well, actually let's not do that.
Let's take this expression and write it without
the radical... the third root... and write
it as the log base four of four to the one
third power. Now in my introduction to logarithms,
I talked about when you log with a base and
you are logging a number with the same base,
that those will cancel out. So this will cancel
out to be 1/3 because the base of the log
and the base of the exponent are the same.
If you have not watched that video though
and you don't know what this is equal to,
then you can write this expression...now I
have made into an equation. I have included
the fact that I don't know what it equals.
I can go from this log form, which I have
scribbled over now, but I am going to take
this and put into exponential form. That is
base, what you get out of a logarithm is an
exponent, and then what are you logging?...
This is the base, exponent, and answer, that
is four to the one third. And when you have
an equation that has the same base on both
sides, that is going to force the exponents
to be the same and there is the same one third
that I got after showing you this cancelation
up here to begin with. When you log something
and the base of the log is the same as the
base of what you are logging they cancel out.
This did not need a calculator. How about
something like, this is a question similar
to a problem that my students had today...
Let's say log base ten of hmmm... trying to
make this up on the fly...log base a 1000
of point 1. What does this equal? We don't
know. Now this could easily be done with a
calculator using the change of base formula
which I am going to get to here but... I want
to show that some of these can be done by
hand. I don't know what this equals so I am
going to put it into exponential form. 1000
to the x power equals .1 If I can get both
sides of this equation to have the same base,
then I can solve for x which is in the exponent.
Let's see, point one is one-tenth right.?
So a 1000 to the x equals 1/10. Well this
is not a fraction and over here I have something
that looks like a fraction, so I am going
to bring the ten out of the denominator by
making the exponent negative. That is going
to be ten to the negative one. Now over here
again we have 1000 to the x. I am just trying
to show every single step. On the right I
have a base of ten and on the left I have
a base of ten... Well can I make a base of
ten? Is there a power of ten in a thousand
so I can get both sides to have the same base?
And yes, this is ten to the third power. Now
we have power to power which means that we
have 10 to the 3x power equals 10 to the negative
one. I am running out of space so i am going
to erase this example here. Now that I have
a like base on both sides of this equation,
I can solve for x and we get 3x equals negative
one. X therefore equals negative one third.
Now we are done. So I was able to solve this
without requiring the change of base formula.
Ok What if you have a question though that
you cannot get the same base on both sides.
That is really where this comes into play.
Let's erase this and show you the difference.
We are going to work out log, actually I am
doing this off top of my head so I am going
to show you how to get this set up so you
can put it into your calculator, log base
two of nine. Ok, now this one we are not going
to be able get set up. If you try to write
into exponential form where it is two to the
x equals nine. There is no power of two in
nine. Like if it were 8, then it would be
x equals three. But no power of two is going
to evenly go into nine...or become nine. So
what are you supposed to do with this? Well
the change of base formula simply says that
if you are logging something you can go "I
don't know how to do log base two." or "My
calculator does not do log base two.", so
I can write common log of 9 divided by common
log of 2. And if you type this into you calculator
properly, you have the answer. Or you can
us Natural Log. This is log base a, they don't
set a base that you have to use on the new
base that you are going to set up.... So you
could us the Natural log of 9 divided by the
Natural log of 2. This is b and this is M,
and now this is this is log base e on the
top and bottom. I have changed from log base
2 to the log base e or base 10. Both of these
can be put into all calculators. If you calculator
makes it easier to use the ln button, do that
log. Well, if you have not studied how to
solve equations with variables in the exponent
yet, this is just sort of doing this because
the equation says so. I want to actually show
you how this comes about. We are going to
do that by going into exponential form that
I just had up here and give you a preview
of how to solve equations with the variable
in the exponent. So what does this equal?
I don't know...supposedly:) So we are going
to go from log form to exponential form. 2
to the x equals 9. But, Mr. Tarrou you just
had that up there a second ago and you told
us that we couldn't do it...well no. The main
purpose or use of logarithms in this level
of mathematics is to solve equations where
the variable is in the exponent. So, if you
are doing compound interest it is very easy
for if you are looking for the amount of time
period...like how long does it take for my
investment to double...that variable is going
to be in the exponent and you have to have
some way to get that variable out of the exponent
so yo can solve for it. You can't solve for
a variable unless it is in the main line of
the equation. You can't solve for a variable
when it is in the denominator and you can't
solve for it when it is in an exponent. So
we are going to apply the log function to
both sides of this equation. You can use natural
log, common log, whatever log you feel like.
I am going to write log, the common log..log
base ten of two to x equals log base ten of
nine. Now does that look funny? It should
not because you have added two to both sides,
divided both sides by five when you are solving
equations, you have square rooted both sides
like x squared equals 16. You applied the
square root to both sides, that is a math
function, so is log. I am just doing something
balanced on both sides, one on each side.
I am applying the log function on each side
of this equation. Well the power property
of logs allows me to take this x and move
it down out front. Now we have x times the
log of two equals the log of nine. How do
you think we are going to solve this for x?
We are going to undo that multiplication and
you are going to see an expression, or an
equation with a part of it that looks just
like uhhh...what we had when I just directly
applied the change of base formula... and
just said Do It because it says so. So we
are going to divide both sides by the log
of two and x is equal to log base ten of 9,
or you could have used natural log, divided
by the log base ten of two. As long as you
type this into your calculator correctly you
will have your final answer. That is it. I
am Mr. Tarrou. I am out. Thank you for watching
very very much. BAM!!!
