All right, so now that we've
looked a little bit at what
a logarithmic
function is and how
it undoes the
exponential function,
we're going to see some
examples in a minute about how
to do a [INAUDIBLE] on
logarithmic function
to solve exponential examples.
But there are also a lot
of properties and rules
that go along with
logarithms, because when
you talk about
logarithms, we're actually
talking about exponents.
So when we look at
something like log base
2 of let's say 64, what the
logarithm is actually doing is,
we're kind of asking a question.
We're saying from
the basis two, what
is the exponent that's going
to give us the 64 that's
of the parentheses.
In other words, two
to what power is 64?
And so logarithms are exponents.
We're trying to
figure out, what's
the exponent in the
base that's there.
And so sometimes
those are more obvious
as we looked at earlier, like
the two to the x equals 8.
If we have this problem
we get something like,
that would be 6.
2 to fourth is 16.
2 to the fifth is 32.
2 to the sixth is 64.
So the logarithm says,
when you have a base of two
down here, what's the
exponent on that that's
going to give you the
value in the parentheses?
So when you have
something like the log
base 10, which is a
pretty common base for us.
Our number system
works in base 10,
so that log base 10 of 1,000--
sometimes these problems
can take a little bit of
work, but sometimes not.
And you think about 10
to what power is 1,000,
hopefully that's pretty
obvious that you get three.
Whereas if you have something
like the log goes 5 of 8,
here again, we're asking
if five is the base, what's
the exponent on 5, meaning it's
5 to what power gives you 8?
That's not an obvious answer.
But that's what this
logarithm is doing.
So that's why it's undoing
the exponents in there.
But because those are exponents,
they have their own rules.
And so you already
know, probably, some
rules about exponents.
And we're going to rewrite
some of those rules using
logarithms.
For example, I'll
give you a couple of--
I'm going to call these
some of the smaller rules.
Some rules that are still
helpful, still important,
but they're not
quite as monumental
of a result as some of the
bigger ones that we could get.
The bigger results
will be on the right.
The smaller ones would
say something like,
the log base b of 1--
and really it doesn't
matter what b is.
I guess I should clarify,
b is supposed to be--
we'll call b a
positive real number.
You could have b
to a certain power.
And number one,
you probably know
that when you take a
positive real number
and you raise to the
0 power, you get 1.
And so this is true
regardless of what
that positive b number you use.
Other kind of
smaller results that,
when you think about what
logarithms are doing,
it's pretty obvious
that log base b of b--
well, again, you're
thinking about b rose
to what power is b, and that
would just be the first.
So other things like the log
base b raised to the x power--
in other words,
what's the exponent
on b that gives you b to the x?
Well, that would just be x.
And so what's interesting
is inside here,
you see there's an exponential
function b to the x,
and then a log based
b of that function.
So you're kind of
composing two functions.
And you may know
from other classes,
or from earlier in
this class, that when
you take two functions that
are inverses of each other,
they undo each other and you
get back the original exponent
that you [INAUDIBLE].
So just kind of smaller
results involving algorithms.
They're still helpful.
They're still important.
And actually, you might
notice that all three of these
are versions of this
third one right here.
The third one is more
important than that,
because the middle one right
here is talking about a case
where you have b to the
first power in parentheses.
The first example
up here is talking
about where you have b to the
zero power, which gives you
one.
So they're all saying
a similar thing.
But some of the bigger rules
can combine some logarithms
together.
And you're going to
think about what you know
about how to combine exponents.
The first one is going
to say that the log base
b of m, plus the log base b of
N. I'm going to take this and--
do we have enough
space here to say this
is the log base b of m times N?
Log base b of m times
N. So this actually
comes directly from something
you know about exponents.
I'm trying to find a way
to label this clearly.
But really this
comes from the fact
that when you multiply
two bases together,
as long as they
have the same base,
you would add their
exponents together.
The second, and bigger
rule, I would say
is almost exactly the
same as the first.
It's just you're going
to change the operations.
So instead of log base b of m--
instead of adding
two logs, we're
going to subtract two logs.
So the log base b of N.
And this is equal to, now
instead of multiplying m and N
together inside
the parentheses, we
are going to take the
log base b of m divided
by N in the parentheses.
And really for pretty much
the same exact reason,
you divide two bases
together, as long
as they're the same base you
would subtract your exponents.
And that's really where
this rule is coming from.
The third rule might
be the most useful,
at least the most used
rule out of these three
by most students.
And it is sometimes called
the power rule, or the power
property, that says the log
base b of m to a power of p.
So you have m to the p inside
of the logarithm, which actually
kind of looks like one of the
smaller rules over on the left.
What happens is you can
bring that exponent of p
down in front.
And a lot of the time
you'll see students
write a little arrow here
that reminds them, oh, I
can drop that down in front.
And this becomes p times
the log base b of m.
And I think this is interesting
because there's not many times
in math where you can kind
of pick something up and say,
I want to move this
piece over here.
And this is one of the only
cases where you can do that.
You say, I just want
to pick up this p
and just kind of move
it down in front.
And that'll make my life easier.
And yet mathematically
that works just fine here.
And where this one comes
from is worth noting as well,
is that when you have a base
that's raised to an exponent
and all of that is raised
to another exponent,
a second exponent--
in other words,
one of the common phrases
that's used in this situation
is thinking about when
you have a power that's
raised to another power,
you multiple those exponents
together.
And that's where you're
multiplying the exponents
together, because you know
the log is an exponent.
And so these rules help
us be able to organize
our logarithms.
They also help us to be able
to give multiple logarithms.
What we're going to
see in the next example
is that third problem, or that
third big rule, that power copy
or that power rule, is
really helpful for us
to be able to solve
exponential equations.
All right, I want to go
back to our finance example,
to our money example talking
about a savings account.
And remember it had a
problem with like, f of t
equals 3,000 times 1 plus .02
divided by 4 raised to the 4t.
And the question was,
when will there be--
was it 5,000?
I think it was 5,000.
When will there be
$5,000 in your account?
And so we could set up
the problem like this.
This is exactly what
we did last time.
And the answer should
look like solving for t.
It should look like being
able to work through.
And what we did first
was we divided by 3,000.
And when we divide
by 3,000 another way
we could reduce that
and put 5/3 on the left,
and this is equal to
1.005 raised to the 4t.
Now, last time what
we did was just
convert it to the
logarithmic form
and have a log with
a base of 1.005.
But what we could do here
is, just like sometimes
when we say let's take the
square root of both sides,
or let's add five to both
sides, in this case we can take
the logarithm of both sides.
And so taking the logarithm
of both sides of the equation
is going to give us something
like the log of 5 over 3,
on the left hand side.
And this is equal
to the logarithm
of 1.005 raised to the
4t on my right hand side.
Now something I
mentioned earlier
is, you'll notice that
there is no base written
right here on this log.
I just said the
log of something.
I didn't put a base down there.
And I probably should have
mentioned this earlier
but when there's a
log like this and you
don't have a base it's kind
of understood that there
is a base of 10 down there.
Our number system
works in base 10,
and base 10 is so
commonly used that when
we have a log with a
base of 10, we just
don't even bother to write 10.
It's just understood
that we're using
base 10 when that's the case.
So now you notice that what
you have on the right hand
side of the equation says
the log of something 1.005,
raised to the 4t.
Here's where those
rules come in handy.
Here's where, especially
that third rule,
that power rule comes
in really handy,
because do you remember what
you can do with this 4 to the t?
When you have a
logarithm out front,
you can take that four and
t and remember we said,
oh, you can just
move that and say,
hey, I want you
to go down front.
And so it does.
And so we have 4 times t
times the log base 10 still.
And it's OK I'm not
writing the base 10.
Just remember that when
there's nothing written,
it's understood that
it's a base of 10.
So we have 4 times t
times the log of 1.005.
And then on the left hand side
we have the log of 5 over 3.
And so we can solve for
t here, and I'm just
going to put it up here
in this top left corner.
You can kind of see
my answer there.
If you work through and you
solve for t, what you can do
is you can divide
both sides by 4,
and divide both sides
by the log 1.005.
And what you're going
to end up with is t
equals the log of
5/3 divided by 4.
And then we also divide
by the log of 1.005.
And you can grab your
calculator and plug these in.
And you want to make
sure that you're
careful about your parentheses.
Please make sure
you put parentheses
around the whole
entire denominator
when you plug that in because
if you don't, your calculator is
not going to give you the right
answer because your calculator
needs order of operations.
So you need to make sure that
the 4 and the log of 1.005
are together in the denominator.
So you can plug this
in to your calculator
and you will get the
log of 5 divided by 3.
And the make sure you
pose those parentheses.
And then divided by--
do you remember I
said you need to make
sure you put that whole
denominator in parentheses.
So the denominator, 4
times the log of 1.005.
And now if you close
both those parentheses,
you've got your whole
denominator in parentheses.
And so you get an
answer like 25.6.
And so think about
what we're [INAUDIBLE]..
Remember we're solving for
t. t, again, is 25.6 years.
The question was,
when will there
be $5,000 in your account.
So it's going to take 25
years for that to happen.
And so when we
have logarithms, we
can bring that t
out of the exponent
and move it down
in front, and it
helps us solve
advanced problems where
you can go through [INAUDIBLE].
