[ Pause ]
>>In this video, we drive
the change of base formula
and we do the following 4
problems and check them.
This is part 14 of logarithms.
We're going to go over what we
call the change of base formula.
Here's the problem.
Let's say I'm trying to figure
out what the log base
2 of 11 is equal to.
There's a little bit
of a problem here
because I can estimate
this answer, right?
I'm thinking 2 to
what exponent is 11?
Well, 2 cubed is 8.
So 3 is too small.
2 to the 4th is 16, too big.
So I know it's 3
point something.
But how am I going to
figure that out, exactly?
We could try to use
a calculator,
except that we don't have
log base 2 on the calculator,
we have log base
10, the common log.
And we have natural log, which
is log base e. So we have
to come up with another
strategy.
Let's say- let's call this
n. We don't know what it is.
So we're looking for what this
log of 11 base 2 is equal to.
Let's just call it
n. Let's change this
from logs to exponential form.
So I have 2 to the n equals 11.
Still a little bit of a problem.
What I'm going to do is
take the log of both sides.
Now we take the log of any base,
the same base on each side.
So let's just do log base 10.
So I'm going to take
the log of 2 to the n,
that will equal to
the log of 11.
Now I could have
used the natural log.
But what I do want
to use is the log
that I have in my calculator.
So I'm going to use
just log for now.
And now, by doing that,
this n in this exponent
can come out to the front.
And that's the key.
Anytime you're trying
to solve something
where the exponent
is a variable,
we want to take the log of
both sides and that allows us
to get that exponent down.
So this gives me n log of
2 equals the log of 11.
And, remember, log of
2 is just a number.
So we're just going to divide
both sides by the log of 2.
And that will be
what n equals to.
So what I really ended
up with is that n,
what I was looking for,
is really just the log
of 11 over the log of 2.
Now notice that is not
the log of 11 halved.
I have to get my calculator
out, use parentheses,
etc. I have to figure out
what the log of 11 is,
figure out what the log
of 2 is and divide it.
This is the exact answer.
That's the exact answer
for the log of 11 base 2.
Now, what if I had done this
problem slightly differently?
Let's say, instead of taking
the log of both sides,
I took the natural
log of both sides.
So let's see what happens there.
I'm just going to take the
natural log of both sides,
then I'll have n
times the natural log
of 2 equals natural log of 11.
And, again, natural log
of 2 is just a number
so I'm dividing both sides
by the natural log of 2.
So this is also equal log of
11 over the natural log of 2.
And, in fact, you can
do the log of any base.
But I'm choosing either
log base 10 or natural log
because I really want to
put this in the calculator
to get an approximation.
So, both of these
are exact answer.
This is an exact answer,
that's an exact answer.
I could do log base 5 of
11, log base 5 over 2.
I'm sorry- the log base 5 of
11 over the log base 5 of 2,
etc. You can have any base,
but I want to use
something in my calculator.
So now we need to put log of
11 in the calculator and log
of 2 in the calculator.
Either both logs base
10 or both natural logs.
So try that in your
calculator and see
if you can get an approximation.
Let's say to nearest-
let's go 4 decimal places.
Now, if you have a
scientific calculator,
usually these keystrokes
at the bottom will work,
whether you use natural
log or log,
just put in the numerator 11 and
then natural log or log divided
by 2 in the natural and
then the equal sign.
And you should get
3.4594, if you rounded it
to the nearest 4 decimal points.
Now, that's sort
of what we guessed.
Remember that?
Because 2 cubed was 8 to the 4th
power is 16, we want to get 11.
So, this seems reasonable.
And, in fact, in your
calculator, if you put 2
to the 3.4594, you will
get a number that is close
to 11, not exactly 11.
Because, remember,
we are rounding here.
You're not going to
get it exactly 11.
So now, let's do
this in general.
So we've got log of
a base b. We want
to know what that's equal to,
so the question is we're trying
to find what it's equal to.
We're looking for n.
So I'm going to write b
to the n equals a.
And just going
to use natural log
on both sides.
In fact, to be really
general, I'm going to say,
I don't care what
base you're using
so let's call it log base m of
b to the n equals log base m
of a. So, in other words, I
would probably just use log,
which is log, or natural log.
But you could use anything.
We bring the n out to the front,
log of b base m. We divide
both sides by log of e base m.
So this n that I'm looking
for actually equals.
Log, now I could use any base,
so that's why you have the m,
base m over the log
of b. So check it out.
It's the log of the a divided
by the log of b. Now, remember,
if you use base 10, you'll just
write log of a over the log
of b. If you use natural
log, it's natural log of a
over natural log of
b. So if you're going
to use your calculator, of
course you would use 1 of those.
And so that is the
change of base formula.
Here it is.
I'm just going to write, log a
over log b; any base you want.
So this is the change
of base formula.
So here are 3 problems
for you to try.
You're going to use this
change of base formula
and you can either use the log
or the natural log key
on your calculator.
So put the video on pause
and approximate each
of these to 4 decimal places.
[ Pause ]
>>All right.
You can either use natural
log, or log base 10.
So I just wrote possible
things you might write down.
And then I did the
computations in my calculator.
Now, you should make sure
that these make sense.
So let's see how you can
possibly check number 1.
So we have log of 20 base 3,
there's approximately 2.7268.
So to check, let's see if 3 to
that exponent is close to 20.
We know it's not going
to be exactly the same.
So now you can use your
calculator to compute 3
to the 2.7268 and
let's see what we get.
I've got 19.99927.
It goes on and on and on.
So, yes. This is 19, sorry.
It's pretty close to 20.
So that looks like it's
a good approximation.
Now, let's do number 2.
So the question is, what
is to the negative 4.1699.
Let's see what happens
when we put this in.
And this is going to be
approximate because I can't go
out all the decimal places.
Be careful that you're
putting in negative 4.1699.
You're going to have to use
that little plus-minus
key in your calculator.
And I get 17.99688, a few more.
So, yes, that looks
like it's about 16.
And finally, let's
check number 3.
So let's do 4 to
the negative 0.1315.
We want to see if that's
going to be about 5/6.
So remember, 5/6 is about
0.8333, so let's see
where you're going to put
that in the calculator.
And using my calculator,
I came up with 0.8333.
And that is close to 5/6.
These are the keystrokes I used.
I put in the 4, that's the base.
Then the y to the x,
to show you're going
to put the exponent- now
you want it negative 0.1315.
So you first have to put 0.1315
and that little plus-minus
key, and then press equal.
And that's how I came
up with the answer.
So just make sure you
practice using your calculator
and that you know how
to check your answers.
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