Today we're going to be talking about logarithmic
and exponential functions.
Actually in the reverse order.
These function are probably, I would guess,
this is an opinion, the second most important
function in the BCS poll of functions with
the first being linear slash polynomial functions
then exponential logarithmic, and then finally
trigonometric functions.
Not that they're the only functions out there,
but they are certainly the most used and the
ones that you see the most often in terms
of applications that includes combinations
of these functions.
Where do exponential functions come from?
We are familiar with linear functions in which
we have a constant rate.
Say that you are earning say 5 dollars and
hour or prices are going up at 5 dollars an
hour, we hope not but let's say they do, those
are linear functions where we would say, predict
here, say we had a starting price of 8 dollars
and every hour they went up by 5 dollars.
We would have this function to talk after
how many hours, say X hours.
After 7 hours what would be the price?
It would 8 dollars, the start, plus the change,
5 times 7 would be 35, we would have 43 dollars
for our final price.
That's a linear function where we have a constant
amount 
rate of change.
Exponential functions though, are those where
there's a constant percentage rate of change.
We would be talking about prices going up
by say, not 5 dollars per hour, but by 5 percent
per hour.
Let's derive what this function would be in
terms of predicting at the Xth hour, how much
would the items cost?
Again let's say they start at 8 dollars today
or right now, and we go up by 5 percent per
hour.
Let's make a table and we're going to have
the number of hours and the price.
It could be the amount in the bank say if
you're investing at 5 percent compounded annually
etcetera.
After 0 hours, we're again going to begin
with 8 dollars.
What happens after 1 hour?
We would have our 8 dollars and we would add
5 percent of 8 dollars, percent increase.
I'm not going to be too concerned with an
actual numerical answer, but rather, as we
often do in algebra, is we're more concerned
about the process so we can do the same process
with very different numbers.
Let's simplify this a little bit.
We have 1 times 8 plus .05 times 8.
We can factor out the 8 and what we're left
with is 1.05.
In other words, we have a 105 percent of the
price that we had before.
105 percent of 8 in this case.
What would happen after 2 hours?
We would begin with our original price, we're
going to write it like that rather than as
just one number, and then we're going to add
another 5 percent of that price that we started
the second hour with.
You can see here we have a common term, this
being 1 times it, of being 8 times 1.05.
That was our common term.
What do we have?
We have another 105 percent of that.
We can see that we can simplify that to be
8 times 1.05 to the second power.
You can see this is going to continue.
After 3 hours, we start with this amount for
the price.
What do we do?
We take 105 percent of that and we get 8 times
1.05 to the third and so on.
At the X hour, we would have 8 times 1.05
to the X power because we've taken 105 percent
X times.
We can see somewhat of a correlation between
this and linear functions.
Linear functions we started out with the rate,
the original amount, and we added multiplying
the rate times the X to get our Y.
In this case we again start with our original
amount, but instead of adding we multiply.
What we see is we get 1 plus our rate to the
X power, 1.05, we would have repeated multiplication,
exponentiation, going on.
Where there was a plus before, with the original
amount, now we have a product, and where we
had our rate before, where there was multiplication
going on, we have repeated multiplication
or exponentiation.
It's almost as if exponential functions are
linear functions, but ramped up one operation.
Let's looks a couple very special exponential
functions, or at least one more.
Say the price, or whatever we're studying
say the amount of bacteria or whatnot, doubles
every hour.
That's pretty easy to see what the exponential
function is going to be.
The number of hours and the price and let's
say again we started with 8 dollars.
After 1 hour it would be 8 times 2.
After 2 hours it would be 8 doubled twice.
After 3 hours it would be 8 doubled 3 times.
In general, in the X case, it would be 8 times,
doubled 3 times, 2 to the third, this would
be 2 to the X power.
Doesn't really look like our previous one
where we had our original amount times 1 plus
R to the N power, or X power I guess we've
been writing.
But if you think about it, it is.
It is going up at a constant percentage rate
because this would be 8 times 1 plus 1, easy
2, to the X power.
What we see is that our rate is just 100 percent.
When we say double, we're saying that the
price goes up by 100 percent rather than say
something like 5 percent every hour.
What about a case where we have the price
tripling every, not every one hour, but every
7 hours.
What would that function look like?
Again we would have the number of hours.
What would the price be?
Again we start with 8 at 0 hours.
All we know is that after every 7 hours, it's
going to be tripling.
We know that at 7 hours, we're going to get
8 times 3.
After 2 times 7, or 14 hours, we're going
to have 8 times 3 times 3, or 3 squared.
What if we wanted to ask, how much can we
predict will be the price say after 4 hours?
We went from 8 times 3 to the 0 power, to
8 times 3 to the first power in 7 hours.
This is four-sevenths of the way.
A good guess would be 8 times 3 to the four-sevenths
power, whatever that might mean.
At least that would lend some credence to
that 3 to the four-sevenths could have some
meaning in that situation.
We see that the exponents, every 7 hours are
going up at a constant rate, somewhat of a
linear function within it just with the exponents.
We should expect the same thing to happen
via the hours.
It's not just the special hours like 7 and
14, but in general we would have 8 times 3
to the X-sevenths hours or dollars for out
F of X.
A little side issue, a lot of people confuse
functions like X cubed versus say 3 to the
X.
These are very different functions.
The reason is, going back to arithmetic, when
we had addition for example, we had 5 plus
4 was the same as 4 plus 5, or 5 times 4 was
the same as 4 times 5 although this takes
a little bit more to argue.
What about 5 to the fourth, versus 4 to the
fifth?
If you calculate these, you're going to find
out that these are very different numbers.
While we had a commutative property of addition
and a commutative property of multiplication,
communicative property brakes down at the
point we would call repeated multiplication
in terms of exponentiation.
X cubed versus 3 to the X as a result of this
arithmetic situation, are two very different
functions.
I encourage you if you want to see the behavior
of these functions, is to set up a little
table, you can do this on your TI-83 calculator,
and you will see that X equal to 0, 0 cubed
is 0, 3 to the 0 is 1, at X equal to 1 we
have 1 cubed, versus 3 to the first which
is 3.
At X equals 2, we'd have 2 cubed which is
8, and 3 squared is 9.
They are different from each other but they
seem to be relatively close to each other.
If you start plugging in values say like 100,
you're going to find a vast difference between
X cubed and 3 to the X.
Both of them blow up, but X cubed blows up
much much slower than 3 to the X.
If you graphed them, you probably could only
see the graph of 3 to the X , on any reasonably
graph, as X gets very very big.
Let's actually talk about the graph of A to
the X in general.
Where A is a positive number.
You might recall if you plotted this graph,
it would be looking like this.
We haven't really talked about negative exponents
or really rational exponents, but what happens
there, say we were looking at Y equal to 2
to the X, here we would be doubling, certainly
as we go left to right, say from 2 to the
first, 2 squared, to 2 cubed, we double in
this way.
In this way we'd be cutting in half.
We would be taking 2 to the negative 1, which
turns out to be a half.
2 to the negative 2 which is one-fourth.
2 to the negative 3 which is one-eighth and
so on.
We will never quite reach 0 as X tends to
negative infinity simply because we'd be taking
a half of a half of a half.
If we did that say with a pizza, we could
keep cutting that pizza in half, and every
time there would still be just a little smidgen
of pizza left.
It's never going to hit the X axis as it tends
to negative infinity.
It certainly blows up as X tends to positive
infinity.
If A is a positive number that less than one,
the reverse happens.
Just for the size of the numbers.
This would be for A bigger than one, and this
would be for A between 0 and 1.
It would tend to look like this you might
recall.
Looking at these graphs in another way though,
say we wanted to solve the problem, 7 to the
X is equal to 38.
Something like that.
This would be saying, at what X value would
we be getting a Y value, using this graph,
of 38.
We would have 38 here and graphically of course
we would go over here.
What you can see is that this graph is a one
to one function.
Any exponential function, at least any pure
exponential function, is going to be one to
one.
That means it's always increasing or, if it's
down here, it's always decreasing s X goes
left to right.
That means that it will have a pure inverse
function.
There is only one number that would solve
7 to the X is equal to 38.
It turns out we give it a really weird notation
from history.
We're kind of stuck with it.
This would be log base 7 of 38.
You would be thinking is this is the exponent
we would be looking at, is what number do
you raise 7 to, to get 38?
In other words, re-writing that equation using
exponentials.
That means that we have an inverse function.
If F of X is equal to A to the X, its inverse
function that would find the Y values, or
the exponent values needed, would be log base
A of X.
It has its own, of course if you look at the
graph reflected about the line Y equals X
of A to the X, this graph would tend to look
like this.
Very very small numbers come from very negative
exponents with 7 to the X for example.
Very larger numbers come from larger X values
with 7 to the X.
We have logarithm functions are the inverse
functions of exponentials.
Exponential grow very very big very fast.
Logarithmic functions grow very very slow.
What number would it take to get say, 343?
A 
sort of large number there, we would find
log base 7 of 343.
It turns out to be a measly 3.
You go way up out to 343, you only needed
an exponent of 3 to go with the 7 to the X.
we would only be up at 3 for our Y coordinate.
If you recall, you also had what we called
rules of exponents: the ways to do operations
on exponents and get an answer fairly quickly
involving other exponents.
You also had analogous rules on logs as well.
For example, we would have A to the X times
A to Y.
What is that?
We say it's A to the X plus Y.
For example we would 7 to the forth times
7 to the fifth.
By definition we would 7 multiplied actually
3 times.
We would have 4 sevens multiplied by itself.
Then we would multiple 7 by itself 5 times.
Combining these, because this is all multiplication
all the time, it's all multiplication, we
would have a total of 9 sevens coming along.
That kind of makes that true for natural numbers.
It's very easy to see that that's going to
be true.
By the way, if we had 7 to the fourth plus
7 to the fifth, don't go making up your own
rules, we would the same set up we had before,
the only problem is we would not have this
multiplication.
We kind of break the chain of the product.
This would be 7 to who knows at this point?
You would have to have another answer but
it would certainly not be a whole number answer.
We would also have say A to the X to the Y
power, that would be A to the XY.
We would multiply it.
Again we could see this, say 7 to the fourth
raised to the fifth power, we would have 7
to the fourth, times 7 to the fourth, times
7 to the fourth, times 7 to the fourth, times
7 to the fourth.
We could break these up into 4 sevens, 4 sevens,
4 sevens, or apply the rule of this from before.
We get a total of 5 groups of 4 sevenths which
turns out to be 7 to the twentieth.
That rule does make sense at least with whole
number exponents.
Finally what about, maybe I should have done
this secondly, A to the X over A to the Y,
is A to the X minus Y.
Again that makes sense.
Say we had 7 to the eighth over 7 to the third,
by definition; this is 8 sevens strung together
by multiplication.
Then down here we have 3 sevens strung together
by multiplication.
We know in terms of canceling fractions, that
we cancel common factors.
We have a whole plethora of common factors
going on.
This 7, this 7, this 7, essentially subtracting
out the 3 sevens going like that.
This would be 7 to the 8 minus 3 or 7 to the
fifth.
What about non whole number exponents?
First of all, what about integer exponents?
What would, say, 7 to the negative 5 mean?
We could use all sorts of different ways of
figuring this out.
One way would be to use rules of exponents.
In fact all of these really: negative exponents,
and we're going to talk about rational and
irrational exponents, basically come from
trying to make sense within the system we
have involving natural number exponents and
just extending that system so that those major
rules of exponents still hold.
What about 7 to the negative 5?
This could be the answer of the reverse of
what we just did, 7 to the third 7 to the
8, which is 7 to the 3 minus 8.
That would be what 7 to the negative 5 would
mean.
By working with fractions and canceling exponents,
this would be 1 over 7 to the fifth.
We would get that familiar rule of negative
exponents become positive exponents if we
just go down stairs with the denominator.
7 to the 0 also would be a common question.
What would 7 to the 0?
There are all sorts of ways to derive this.
But we could take 7 cubed over 7 cubed, by
that rule of exponents if we wanted it still
to hold, then 7 to the 0 better be equal to
a number over itself which is equal to 1.
Any number to the 0, any positive number I
should say to the 0, is equal to 1.
Funny case of 0 to the 0 is an indeterminate
form.
In one case we have 0 to a power which should
mean a 0, and the other case is something
to the 0, which means the answer is 1.
Which one do you choose?
Both of them are just as plausible, just as
viable, and actually this could end up being
anywhere in between.
You'll be studying things about indeterminate
forms as we go along through the semester.
What about rational exponents?
What about say, X to the one-fourth?
What would that mean?
What would like to do is say, what about,
what would we have, whatever X to the fourth
might mean, it should fit the rules of exponents.
If we take X to the one fourth and multiply
it by itself, we have four of them stung together
by multiplication, by rules of exponents we
should be able to add those exponents and
get X to the first power.
What should X to the one-fourth mean?
That's the number you would multiply by itself,
you would have 4 copies of it multiplied together
to give you the number.
7 to the one fourth is a number that you would
raise to the fourth power to get 7.
Basically, rational exponents are roots.
This would actually be called the fourth root
of 7.
What about 7 to the five-fourths?
Again invoking rules of exponents this would
be that fourth root of 7, 5 times.
Multiply by itself, a string of them multiplied.
Finally what about irrational exponents?
What about 7 to the pi value?
What would that mean?
We can't really split this up with the rules
of exponents very well with the irrational
number like we could with rational numbers.
What we could do is say, there are a ton of
rational numbers just sitting around pi that
we could start finding out, what about 7 to
the third?
3 is very near pi.
What about 7 to the 3.1?
That's getting closer to pi. 7 to the 3.14.
7 to the 3.141.
We could keep going, keep estimating pi and
getting closer and closer to pi.
It turns out, how we define with an irrational
number is somewhat of what we'll call a limit
process.
We're going to basically say it's the limit
as X gets closer and closer to pi of 7 to
the X.
We can use rational numbers to try to get
a picture or a feeling for what that might
mean since we already know how rational exponents,
say work with fractions and roots.
Looking from the logarithm side, we had all
those rules of exponents: A to the X times
A to the Y is A to the X plus Y.
A to the X to the Y, let's just look at these
two, would be equal to A to the XY.
Logarithms have analogous rules.
If we had log base A of XY, this one will
be very analogous of this.
It turns out to be log base A of X plus log
base A of Y.
What about log base A of X to the Y?
That turns out to be as if you flipped this
over.
Log base A of X, taking Y copies of that.
That kind of makes sense.
You could derive it using the rules of exponents
but even makes sense say, log base A 
of 3 to the fifth power, you're basically
saying what number do you raise 3 to get 3
to the fifth?
It's almost like saying, what color is the
green grass?
We are basically treating this is the inverse
function.
Log base 3, we exponentiate first, take the
log, we're right back to the original place
at the 5.
That's a very special case of that and because
of this rules of exponents, you can re-derive
this to be not only for things of base A,
but for any X of Y that it works.
From these two rules of exponents, you will
be using these rules big time in something
later on called logarithmic differentiation.
For now it's just good to visit an old friend.
I guess I should have gone by this rule.
This is 5 log base 3 of 3.
Again, what number do you raise 3 to get 3?
5 times 1 which gives you the 5.
Sorry about that on that last example.
If you want we can add in another, A to the
X over A to the Y is A to the X minus Y.
Analogous rule of log, would be log base A
of X over Y, would be log base A of X minus
log base A of Y.
Just to wrap things up, we have log of a quotient
is the difference of the log, just like log
of the product was the sum of the logs.
That is all.
