Welcome, ladies and gentlemen.
So what I'd like
to do is show you
how to convert between
exponential and logarithmic
equations.
And this is a very
important process
to not only
understanding logarithm,
but also once we
get into solving
exponential and
logarithmic equations.
Going in between the two
equations can be very helpful
and, I think, really,
really important as far
as understanding
what a logarithm is,
which a lot of students
have trouble with.
So logarithmic in
exponential functions
are what we call
inverses of one another.
And the way that they're related
is if we look at this equation,
you have 3 squared equals 9.
Well basically, all
we're simply going
to do to rewrite this-- we can
work the slow step by step,
and that's what I'll do
for the first problem.
It's just really break it down.
And you can see I have
an exponential equation.
And then I have converted
that exponential equation
in logarithmic form.
So you can see where the Y's
go, the B go, and the X's go,
where y, b, and x all
represent real numbers.
So if you're having
a little trouble,
or you're just in the start, we
can just write everything out
as far as what they represent.
So we can say b,
in this case, is 3.
X is equal to 2.
And y is equal to 9.
All right?
And the reason why
I'm doing 12 examples
is because the more
and more you do this,
the faster and
quicker and easier
it is for you to understand.
All right, well now
I want to convert
this a logarithmic form.
So all I'm simply going to do is
take my values of b, x, and y,
which I got from this
form of exponential.
And I'm going to rewrite
them in the log form.
So it's going to be the
log, and then base, b.
So that's b, which is in
this case 3 rate of y, which
in this case is 9, is equal
to x, which in this case is 2.
OK?
So when we're
looking f logarithm--
and I'll just say this.
What a logarithm basically
is stating-- this
will become important once we're
trying to convert logarithms.
The logarithm is
basically saying,
the log base 3 of
what value is 9?
Well, that value is 2.
So 3 raised to what power is 9?
2, the answer is 2.
OK?
So a lot of times, you can
always check your work, right?
You want to make sure
that whatever the base is
of your logarithm
raised to the solution
as a power is going to
equal your value of you
in your logarithm.
The next thing I also
like to point out,
which really helps me understand
this, is b represents the base.
Here you have b as the
base of the exponent.
Well in logarithms,
we have bases as well,
and-- which actually, I
didn't do any of those.
I'm going to have to
add a couple of those.
OK.
So in this case, we
have base-- notice
how b is always the base.
So when you're converting from
exponential to logarithmic,
you're always going
to have that base 3.
Yeah, I want to do a base 10
and a base e for you as well.
So I'll get to those as well.
OK.
All right, so let's get to it.
So therefore, in
this case, we'll
just move through
these pretty quickly.
Here's going to
be, again, my base.
So it's just going to be log of
base 2, and then of 8 equals 3.
So over here, my base is 1/4.
So it's log of 1/4
of 1/16 equals 2.
And again, you can
always check your work.
1/4 squared, is that
going to equal 1/16?
Yes, that does work.
And then over here, now my
base is on the right hand side.
But that's OK.
It doesn't have
to be-- actually,
it can look just like this.
That's OK.
You notice your base of your
exponents over here, too.
So it's going to be log base
2 of 1/16 equals negative 4.
Over here, my base,
again, is over here.
So it's going to log
base 5 of 625 equals 4.
And again, does 5 raised to
the fourth power equal 625?
It does.
Now this one's a
little bit difficult
because we don't have a
base or an exponent, right?
So kind of threw in
a little trick here.
Well remember, we can
always rewrite radicals
as rational powers.
Now, by rewriting
in this format,
we can see that 27 is my base.
27 is my base of 3 equals 1/3.
OK?
So now we're going to be
doing is converting back,
going from-- so that's
how we go from exponential
to logarithmic.
Now what we're going
to do is go from
logarithmic to exponential.
And remember what I spoke about.
A logarithm is 3 raised to what
power gives you your value 9?
Well, that answer is 2.
So we know this is going
to be 3 squared equals 9.
So again, my main thing,
though, I always like to look at
is look at the base, right?
The base is 3.
So I know the base
of my exponent is 3.
And we should
know-- hopefully it
doesn't make sense that 3
to the ninth power equals 2.
Right?
That's not going to
be your exponent.
3 to the ninth power is not 2.
So it's going to be
3 squared equals 9.
Over here, my base,
my logarithm is 2.
So that's going to be 2 to
the fourth power equal 16.
Over here, I have 3 to the--
my base of my logarithm is 3,
so that's going to be
the base of my exponent.
3 to the fourth power equals 81.
Over here, I have 16 to
the 3/4 power equals 8.
Over here, my base
of my logarithm is 7,
so that's the base
of my exponent.
It's going to be 7 to
the negative second power
equals 1 over 49.
And again, just notice I'm
doing the same product.
It doesn't matter if I have
negative and rational numbers,
or if I'm just doing this.
It's 3 to the second
power equals 9.
7 to the negative
second power equals 149.
It's the exact same
process over here.
This one's going to
be the base, is 1/3
raised to negative 3 equals 9.
OK.
So I quickly went
through all of those.
But there are a couple problems
that I want to go through
that I didn't write down.
I actually just forgot.
So I'm going to do
four problems for you.
Yeah, I'll do four
problems for you.
10 to the second
power equals 100.
That's why I didn't do e,
because I didn't know my e.
OK, well actually, let
me make up a problem.
e to the third power,
so approximately 20.08.
Let's do log of 7.
And let's do ln of 5.
OK.
So the reason I want to do these
is because these deal with base
10 as well as base e.
And these are going to be
your most commonly used bases
in logarithms.
And they're also the two
bases that your calculator
is going to use.
So when you plug-in
log in your calculator,
it's only using base 10.
So therefore, we can't
evaluate the log base 3 of 81.
You have to use the
change based formula,
or make sure your
calculator has the ability
to change different
bases other than base 10.
But most calculators
only evaluate in base 10,
so you can just use a change
of base formula, which
we'll go over as well.
And base e is going to be
your natural logarithm, which
is represented in
your calculator as ln.
So anyways, the main important
thing I wanted just to go
through-- again, we're going
to follow the same process.
This is log base
10 of 100 equals 2.
But the main important thing
which you'll see very common
is we don't really
write log base
10, because that's kind
of like our default
base for a logarithm.
So we'll just write
log of 100 equals 2.
And I wanted to
do these problems
because I just wanted
you to understand
that 10 is still there.
It's like writing the number
8 knowing that there's a 1
in front of it, right?
So the base 10 is assumed when
no other base is provided.
The same thing for e.
We'll just write this
as log base e of 20.08.
OK?
And I'm using
approximate because e
is an irrational number.
So doing logarithms
with irrational numbers
is going to still
give you approximates.
However, for log base e, we
use the natural logarithm,
which is the exact same.
It's just our notation is
going to be-- we just used
the natural logarithm.
And again, when
we're using ln, We
know that the base
is going to be e.
So we don't write in base e.
And you'll see it on a
test or in a textbook
like that, where the e is
not written as the base.
It is assumed that you'll know.
So when a lot of times
this comes into problems,
they're these two
problems, because students
will look at these and say, I
don't know what the base is.
Oh shoot.
Actually let me change this.
Oops, I forgot to
finish the problems.
OK.
OK, forgot to write that in.
So now, a lot of
students are like, well,
I don't know what the base is.
Well remember, logarithm, if
there is no base provided,
we use it as base 10.
So you're just going to write
in a nice little 10 there.
And for the ln, when
the base is not given,
we just put a nice
little e there.
So now, I can say 10 to the
negative first power equals
1/10.
And e to the 1.6
is approximately 5.
OK?
So there you go
ladies and gentlemen.
That is just a quick
little overview
of how to convert between
exponential and logarithmic
equations.
Thanks.
