Hello, and welcome to
Bay College's video
lectures for
Intermediate Algebra.
This is Section 8.6,
which introduces
common and natural logarithms.
The first one we're
going to look at
is common logs,
which means base 10.
And the reason why
this is the common log
is because our number system
is based on factors of 10.
If we look at the decimal
point of our number system,
anything to the left
of it is a factor of 10
compared to anything
to the right of it.
So when we deal
with common logs,
it's so common because our
number system is based off
of a base 10 number system.
So when we see a log
base 10 of some argument,
it's actually equal to
log of the argument.
It's so common that we
do not denote a base.
So whenever you see a
logarithm that does not
indicate a base, where the
space for a base is blank,
you have to assume that
this is a common log.
It is a base 10 logarithm.
So let's look at
an example here.
We have log of 100.
Well, being that the base isn't
indicated, it is a base 10.
So what does the log ask?
It asks 10 to what power is 100?
Well, I know 10 squared is 100.
So this value is 2.
10 squared is 100.
So log of 100 is 2.
What about log of 1,000?
Well, this is a factor
of 10, 10 to the third.
So log of 1,000, without
the indication of a base,
I assume 10.
10 to the third is log of 1,000.
Here it's a little tricky.
But as I said, our number system
is based on factors of 10.
This value here, 0.01, is
the same thing as 1/100.
These two values are
actually read the same way.
So if I have a decimal,
I know that I'm
going to have a
negative exponent
because the factor of 10
is in a denominator, 1/100.
So this would be 10 squared.
In the denominator, it would
have a power of negative 2.
10 to the negative
second is 1/100.
And this is 1/100.
So the value here
would be negative 2.
Now, what if we're dealing
with common logs base 10,
and they don't have a nice
factor of 10 in the argument
here.
Well, let's look at this here.
We have log of 500.
In the previous
section, we explored
the rules of logarithms.
And if we truly want to think
of why we are learning this,
well, 60 years ago we
didn't have calculators.
They weren't accessible
to the public.
Computers were-- they
occupied entire rooms.
They didn't have
personal computers,
let alone calculators.
So what engineers who
generally worked with these,
and in other fields as well,
they had a book of logs.
But it couldn't list
every single logarithm.
And since they didn't
have a calculator,
they used those
properties of logarithms
to break this down
into something smaller.
So I'm going to do that as
an example for this one here.
I have log of 500, which is
not a perfect factor of 10.
But I know the base is base
10 because it's a common log.
It doesn't indicate the base.
So I'm going to rewrite this
using my properties of log.
I know that 500 is 100 times 5.
Using the product rule of
logs, I can split it up,
log base 10 of 100
plus log base 10 of 5.
So log of 100 plus
log of 5, this
is something I can
evaluate-- 10 squared,
just like we saw on
the previous example.
So this is 2 plus
whatever this value is.
So I could refer to
my book of logarithms
and find the simple
logarithm of log 5.
And I find that it
is-- and I already
cranked into a
calculator because we
do have that luxury these
days-- this would be 2.69897.
And this is an
approximation because we
have the log of 100, which
is 2, plus the log of 5.
Well, this value is less
than one factor of 10.
This value is less
than 1, 0.69897.
So when we put the two
together, we get 2.69897.
Now at this point,
I'd like you to have
a calculator
available to you as we
go through the
rest of the video.
Plug this value in,
log of 500, and you'll
get this value to
so many decimals.
It is an approximation
because even our calculators
have to round off that
decimal to some point.
But we're going
to five decimals.
Well, what if I had log of 242?
Well, if I do an
initial assessment,
I know this value, because
we're dealing with base 10,
is somewhere between
100 and 1,000,
those nice whole
integer powers of 10.
So if I plug this
into a calculator,
one way to know that
I'm on the right track
is to know that
whatever the value is
should be somewhere between
2 and 3, the powers of 10.
10 squared would be 100.
10 cubed would be 1,000.
This value is somewhere
between there,
so somewhere between
the powers of 2 and 3.
When I do plug this into a
calculator, I get 2.38382.
0.38382 And there'd be other
decimals in our calculator.
But, again, we're going to
five decimals, so 2.38382.
To check your work with a
logarithm, take the base of 10,
raise it to this power.
And because we had to estimate,
you'll get really close to 242.
What about this value here?
Hopefully, we don't
even need a calculator.
But I do want you to put
it into a calculator.
And if you do that right now,
hopefully your calculator
tells you that this value is 0.
Because regardless
of the base-- this
is one of our rules of
logarithms-- the log of 1,
regardless of what that
base is, is always 0
because any base to
the 0 power is 1.
Now, put this into
your calculator.
And depending on the type
of calculator you have,
it should tell you
something about this value.
This is not a value.
Log of 0, we can't
take the log of 0.
It's a domain error.
So maybe your calculator
says domain error.
Or maybe it just says error.
It really depends
on your calculator.
You might get this message
when you plug that in.
So plug it in.
See if you get that message
or something similar to it.
It might say error
domain restriction.
Also, this value here.
Because when we talked about
exponential and logarithmic
functions, we knew a little
bit about their domain,
the log of a negative.
Well, you can't take
the log of a negative.
But you can try to punch
it into your calculator.
And you may get something
that either says error.
Or it might even say
non-real solution.
So this is not real.
And for intermediate
algebra, that's
as far as we're going to go
with logarithms of negatives.
This is not a real value.
It's not within the
domain to be able to take
the log of a negative value.
Let's look at some
examples and use some tools
that we learned before.
Now, here I have log of x.
And it's base 10.
Because it doesn't
indicate that,
it is a common log-- equals 2.2.
Well, if I were to
solve this equation,
the argument is my variable.
So I'm going to rewrite this
as an exponential equation.
I identify the base to
be 10, a common log.
And this is the power, 2.2.
This is something I can
plug into my calculator.
But because my calculator
is going to round it,
I know it's an approximation.
When I put this into a
calculator, I get 158 point--
and we'll go to four
decimals-- 4893.
So if we think about this,
I have the base of 10.
And I'm raising it to a
little bit more than 2.
Well, 10 squared is 100.
This value should
be more than 100.
And we see that it is.
So we can estimate those values
instead of relying solely
on that calculator.
Well here, we have
in this next example,
we have 10 to the x
power equals 26.4.
Well, if I'm going to
estimate a solution,
this power has to be more
than 1 but less than 2
because 10 to the first is 10.
But 10 squared is 100.
This value is somewhere
between 10 and 100.
So this power has to be
somewhere between 1 and 2.
So to plug this
into a calculator
to find an approximate solution,
or at least more approximate
than between 1 and 2, we
can rewrite this equation
as a logarithm.
Log base 10, which
is a common log,
so I don't have to indicate
that, of the argument 26.4.
This is something I can
plug into a calculator.
And when I hit my Log
key of my calculator,
and I put in this
value, depending
on the type of
calculator you have,
you might have to put
in the argument first
and then hit the Log key.
It really depends
on your calculator.
We get an approximate
value of 1.4216.
So I'm only going
to four decimals.
Your calculator carries
it out a few more.
But if you round it
to four decimals,
this is the approximate
value you would get.
So what I want you to do is
very similar to this one.
Rewrite this as a logarithmic
equation, and solve for x.
So try that one on your own.
Let's look at
another type of log.
And it's called the natural log.
This is of the base e.
And we touched on
the natural number
in a previous video
for this chapter.
But e is nothing more
than an irrational number.
If we think about the
value of pi, some of us
are very familiar with pi.
We know that that's 3.14.
Maybe we know it to a few
other significant figures.
It's 3.141592654.
And even that would continue.
So if we're familiar with pi, we
should become familiar with e.
e is just a symbol that
represents this irrational
number, 2.7182818284590.
And this value would
continue on and not
have a repeating decimal.
It doesn't terminate.
So if we think about e, it's
just a number similar to pi.
Except it's a different value.
So we just use this symbol.
On our calculators, we might
see this value here, e to the x.
This is what we're going
to use in our calculators
because we have
that convenience.
And when we talk
about the natural log,
we should see this button
right underneath it.
So you might have to hit Shift
to access e to some power.
But the ln key is
a key on there.
Now, it's called
the natural log.
So you might think
it would be nl.
But because many of our
mathematicians in the 1600s
were of the country of France,
the Latin-based languages
speak a little differently.
This is log naturel.
So natural log, log
naturel, that's why it's ln.
That's the symbol we use
to indicate log of base e.
So if we have log base e of
x, our shorthand notation
is ln of x.
This indicates base e.
The base is this
irrational number.
Now, we might
think, well, why do
we have this irrational number?
Why do we have a specific log
function on our calculators
for this number?
Well, this number
appears in nature.
It appears when we talk about
exponential growth of things
like bacteria.
We see it in
business when we talk
about compounding interest.
And that's something that we're
going to look at as an example
before the end of this video.
So let's look at ln of 4.
Let's find this approximation.
We're going to use
our calculators.
We're going to hit that ln
key and put in the value of 4.
And if we think about
e, e is 2.7 something.
Well, if I raise 2.7 to the
first power, I would have 2.7.
So I know this value is
going to be more than 1.
But it may be less than 2.
So it's going to be a
relatively small value.
And if I do put this
into a calculator,
I'm going to get 1.38629.
1.38629, and this is an
approximation to five decimals.
So plug this into
your calculator.
And you should get
a similar value.
Know where that function
key is for a natural log
so that you'll be
able to utilize it.
The next example we're going
to look at is ln of 18.
Now, if e, being
an rational number,
can sometimes be a little bit
more difficult to estimate
than base 10, but if I
look at this and say,
well, e is a value close to 3.
And 3 squared would be 3.
And 3 cubed would be 27.
So powers of 2 and 3 would
get me close to this value.
So I'm going to assume
that my value is somewhere
between 2 and 3.
So when I put this
into a calculator,
I'm going to take the
natural log of 18.
And my approximate value
is going to be 2.89037.
So we can see that
approximation.
Type that into your calculator.
Make sure you get that value.
But realize that you should
be able to estimate it.
Let's look at this here.
And I'm going to
put an equal sign.
If you put this value in your
calculator-- and hopefully
you're following along
and doing that-- you
will get the value of 1.
This is equal to 1.
Because what does a log ask us?
Well, regardless
of the base, it's
saying the base to what
power is the argument?
e is our base here. e
to what power is itself?
e to the first power is e.
So we get 1.
What if we put in this value?
Well, if we put this value
into a calculator, we get 0.
And hopefully we recall one of
our properties of a logarithm
is the log, regardless of its
base, if it's argument is 1,
is always equal to 0 because
anything to the 0 power is 1.
e to the 0 power is 1.
It holds true.
And, again, we have ln of 0.
If we put that into
our calculator,
it's going to tell us there is
an error, domain restriction.
If we put in negative 4, we're
going to get a similar message.
Maybe it says error, or
it says not a real value.
So be aware of that.
You can't take the
log of 0 or the log
of a negative,
regardless of their base.
Just like we saw with base
10, we see that with base
e, as well.
So let's look at
some examples where
we might have to solve a
logarithm or exponential
equation with a base of
e. ln tells me base e.
So I know the base.
I know the power.
This is the argument.
So I can rewrite this equation
into exponential form.
e to the fifth power is x.
Now, this is
something that I can
plug into my calculator,
which I've already done.
And I got 148.4132.
Now this is an
approximate value for x
because I had to round
it at some point.
The calculator may carry
it to a few more decimals.
But it still has to round
it off at some point.
So we'll go to four decimals.
This would be the
approximate value of that x.
And we can check our work.
If I go back to my calculator
and take the ln of this value,
it will tell me 5.
What if we have this example?
We have an equation where
we have the base of e
to the x power equals 25.2.
Well, I can rewrite this as
a logarithmic equation, ln
of the argument 25.2 equals x.
Now this is something
I can simply
plug into my
calculator, ln 25.2.
And I'm going to get an
approximate value of 3.2268.
So we see we have this value.
And if we think about if, e is
a value almost 3, but not quite.
And if we think 3
cubed would give me 28,
this number is
pretty close to 28.
Obviously this is
a little bit more
because our base was
a little less than 3.
Now, e to the x equals 7.
I want you to do
this one on your own.
Rewrite it as a logarithm,
and solve for that power.
Now we're going to look at
an application of where we're
going to see this will lead
into e appearing in nature here.
So we have compound interest.
In this application, you
can see we have exponents.
And when it comes to solving
exponents, we use logarithms.
So this is a related
example here.
For compound interest, we have
to define what these terms mean
for our compound
interest equation.
You may or may not be
expected by your instructor
to memorize this.
But what we have here for
compound interest, something
we use very common in business
or in banking or anything
like that.
Maybe you purchase a home,
or you take out a loan,
or maybe you have
a savings account.
It's going to follow
compound interest
or something similar to it.
So if we define the
values we have here,
A is the total end
amount after some time
of being in an account or
gaining interest at some point.
P is the principal,
how much money
we're initially starting with.
r is the annual interest rate.
And hopefully we recall, if
we're given rate as a percent,
we have to transform
that to a number,
move that decimal place.
n is the number of times
of compounding per year.
Now, it may say annually,
which means one compounding.
It might say semi-annually,
which means two.
It might say quarterly,
which means every three
months, so four times a year.
It might say monthly or daily
or biweekly, 26 times a year,
or weekly, 52 times a year.
So we have to watch
the terminology
when we come to
application problems
because the number
of compoundings
might not be clear.
It might be given in words
such as semi or quarterly
or something like that.
And t indicates the
number of years.
So of these variables, A is the
amount after-- the end amount
after some time.
P is the initial
amount, what we put in.
And we have 1 plus the interest
rate over the number of times
per year we're going
to compound it.
And what it means
to compound is we're
going to add interest to it
before the end of the year.
And, again, we have n, which
is that same variable here,
the compoundings
times the time, t.
So let's look at an
application here.
I want to know the
amount of money
if I invest $100 at a rate of
2.5% compounded semiannually
for 10 years.
So I'm going to put $100
into an account that's
earning 2.5% interest.
And every six months,
semi-annually,
twice a year, they're
going to take my interest
and add it to my principal,
so I can earn interest
on my interest.
That's good news for me.
And I'm going to do
that for 10 years.
So I'm going to
use the equation,
the amount equals
the principal, which
is $100, times the quantity 1
plus the interest rate, 2.5%,
is 0.025, over
semi-annually, that
means n equals 2,
twice a year, raised
to the nt, which is n
times t, 2 times 10.
And if you wanted to
simplify that, you could say,
well, that's just 20.
Now, here's a value.
I could simplify this.
I could add 1 and then
raise it to the 20th power.
Well, since we do have the
benefit of calculators,
this would be a good
time to use one.
And we're going to find that if
I take 1 plus 0.0125 and raise
it to the 20th power
and multiply it by 100--
be sure you're following
order of operations--
I'm going to get an
amount of 128.20.
And that decimal's
going to continue.
But, since we're dealing with
money, our initial investment,
we're going to round
it to the 1/100
because that's our
monetary unit here.
We go to the penny.
So we get 128.2, or $128.20.
Now, what would happen if
I changed my compounding
because I want to earn
interest on my interest?
What if I said, well, let's
compound it quarterly?
Well, that would be four times.
Or I say monthly, that
would be 12 times.
Or I'd say weekly, 52 times.
Well, what if I went daily?
Could this earn me so much
money that I'd be filthy rich?
Well, that'd be awesome.
But what we find--
what happens is
let's go back to this
original equation here.
If we compounded it an
infinite number of times,
if we added the
interest at every moment
that we were gaining interest,
what happens to this equation?
Well, n is getting infinitely
large because we're
going to compound it over
and over and over and over.
Well, if I divide a
value by a large number,
it gets smaller and smaller.
So this becomes 1
plus a value that's
getting closer and closer to 0.
1 plus almost 0 is
almost more than 1.
It's 1 and almost
a little bit more.
But I'm raising it to
the infinite power.
Well, 1 to any power is still 1.
And since this value is
getting closer and closer
to just being 1, 1
to the infinite power
actually becomes this equation
when n goes to infinity.
And this is something
that's called a limit.
And that'll be introduced
in the next math class.
But what we get is A
equals P e to the rt.
This value actually becomes
e, a fixed constant.
Right?
Even though it's an irrational
number, it's just a number.
So it wouldn't result in
making me all this money
because it has a limit.
And that limit is
the natural number.
So if you wanted to,
just to see what happens,
do this equation with,
let's say, $1 here.
And keep raising n to the
highest and highest and highest
power, and see that that value
will eventually become e.
So let's look at continuous
compounding interest.
Here we have essentially
the same example.
But we're going to let
it compound continuously,
that means we're letting
that n go to infinity.
So we don't need n here
because it decayed to e.
It has a limit of e.
So the equation for
continuous compounding
is A equals P e
raised to the rt.
So if I want to know
how much more money do
I get if I compound
it infinitely,
well, if I invest $100 at
2.5% for 10 years compounded
continuously, n is infinity,
I use this equation here.
A equals 100 e
raised to the 2.5%,
which is the 0.025,
times the time, 10 years.
So I'm going to put it
into the same account,
or the same interest for
the same amount of time,
but continuously compounded.
And if I plug this
into a calculator--
and make sure you follow
your order of operations--
I'm going to get $128.40.
Well, even though I
compounded it infinitely,
I only made 20 more cents.
So it does have its limit.
So $128.40, well,
yeah, that's more,
if I compounded it infinitely.
But it's not a
significant amount
because 2.71828,
that value of e,
is not a value that
much greater than 1.
So we earned that much interest.
So this has been an application
of continuous interest
and compounded interest.
This has been Section 8.6.
Thank you for watching.
