Hey everybody this is video of the week
number 11 this is just gonna be a really
brief introduction to exponential
functions and two logarithms. Now here's
an example of an exponential function Y
equals 2 to the power of X. If you
wanted to investigate what that graph
might look like you might start with a
little table of X and Y values and maybe
I'll make up a few X values right here
to plug into our equations. Don't forget
that if you plug in X equals zero you
get 2 to the zero and anything to the
zero power is just one. If you plug in X
equals 1 you get 2 to the first
power if you plug in X equals 2 you
get 2 to the second power which is
4 and if you plug in X equals 3
you get two to the 3rd power which is
eight if you plug in a negative number
for X you get a negative exponent you
might have to go back and review your
negative exponents but if you plug in X
equals negative one you get 1/2 and if
you plug in X equals negative 2 you're
gonna get 1/4. Now we can plot all of
these points on our XY plane over here
they're gonna look something like this
and if we connect all of our dots with a
curve our exponential function ends up
looking something like this. This is a
horizontal asymptote that this function
approaches at Y equals zero and there's
not a vertical asymptote over here. You
might ask the question what does Y
equals...4 to the X look like and the
answer is it looks pretty similar it's
just that it grows a whole lot faster.
It's also important to point out that
there's a very special exponential graph
in between these 2. Y equals E to the X
where E is some irrational number just
like pie is it's a number that goes on
forever without repeating 2.71828 and so
on. It's worth noting that all basic
exponential functions pass through that
same point zero one. Of course in this
chapter we're gonna take these
exponential functions and shift them up
and down and left and right and after
you do that they'll no longer pass
through zero one but the basic
exponential functions do. That of course
is because if you plug in X equals zero
into any of these exponential functions
you get something to the zero power and
that is always 1. Okay in this chapter
you're gonna learn about inverses and
most students don't struggle too much
with inverses so I just want to talk
about the inverse of exponential
functions. Now the inverse of exponential
functions are called logarithms and
we're going to talk about these for the
rest of the video. I'll start by showing
you a graph of what a logarithm looks
like. Now you learned that if we graph
the inverse of a function
it's like reflecting that function over
the line Y equals X right here or in
other words taking all of these points
and switching the X and Y values. So if
we take Y equals 2 to the X and switch
all of the X values over the graph looks
something like this and what we do is we
call this y equals log base 2 of X. We
could do the same thing with a log base
4 of X and our very special log base E
of X. All we did was reflect the graphs
above across the line Y equals X. We took
those graphs right there we reflected
them that means that they're an inverse
and we're naming these things log base 2
of X log base E of X and log base 4 of X.
Now that horizontal asymptote Y equals zero from above is now a vertical asymptote X
equals zero and there is no horizontal
asymptote here. Now this log base E of X
is such a special function that we give
it a different name we call it the
natural log of X or log natural X. It's
just that we happen to deal with this
function this log base E function so
often that we want to kind of shorten
the name. Okay so we just learned that
exponential functions and logarithms are
inverses really what is a logarithm.
Let's think about that question by
looking at a specific exponential
function. Y equals 2 to the X power. If we
think about taking an inverse of that
exponential function what we can do is
we can just switch the X and Y values
and to find our inverse of our original
function in previous sections what we
would do then is we would solve for Y. Of
course we don't really actually have a
way to solve for Y here unless we invent
something new. So the new thing that we
invented is a logarithm. If we solve this
equation right here for Y we get Y
equals the log base 2 of X. Okay now this
piece right here is actually quite
important. So why don't we shed a little
bit more light on what we're talking
about by picking some very specific
numbers. I'm going to plug in X equals 16.
Now these 2 equations are exactly the
same thing the 16 equals 2 to the Y and
this Y equals log base 2 of 16. So what
you can recognize by looking at these
two equations is that this Y this
logarithm right here is actually just a power. You could
probably come up with in your head what
this Y should be. Two to what power
equals 16, well if you thought about
it you would get that Y should be 4.
Likewise this number right here should
be 4 so the 2 equations that we
have now that are exactly the same thing
look like this. Sixteen equals 2 the 4th
power is the same thing as saying 4 is
the log base 2 of 16. So what if we just
learned if we look at this equation
right here what we learned is that the
log base 2 of 16 is the power on 2 that
gives us 16 and that is 4. So as a quick
example what would the log base 10 of
100 be. It's the power on 10 that would
give us 100, so 10 to what power is 100
think about it for a while and the
answer is going to be 2. As another
example let's say what is the log base 3
of 81. In other words what is the power
on 3 that would give us 81. Well you
might have to write it off on the side
how many times would you have to
multiply 3 by itself to get 81 and the
answer is 4, 3 to the 4th power is 81.
Now these can get a little trickier of
course something like log base 2 of 1/4
what power on 2 would give us 1/4. You can think about it for a minute by
hitting pause on your browser.
I'm gonna scroll back up to our very
first example we're asking 2 to what
power gives us 1/4. If we look up at our
original graph right at the beginning
here we wrote down that 2 to the
negative 2 power gives us 1/4 right
there. So the answer to the problem down
here...is just going to be negative 2, 2 to the
negative 2 power is 1/4. Okay remember up here when we were writing equations in
2 different forms this looked like an
exponential equation and this was an
equation with the logarithm n it. If we
investigate that a little bit more
generally what we get is this. If we have
a logarithmic equation Y equals log base
a of X it can be written as an
exponential equation as well. A to the Y
equals X, and we can take any exponential
equation
and write it as a logarithmic equation
just by moving the letters around. So
just as an example if you have yourself
this logarithmic equation M equals log
base 2 of 17 we can take those letters
and move them around appropriately based
on this formula right here and you would
get the basis to power on 2 is M and
the right side of this equation here is
going to have to be 17. We're saying that
M is the power on 2 that gives us 17.
Likewise if we had this equation right
here in exponential form we could write
it as a logarithmic equation right here.
Okay now logarithms have a whole lot of
properties they're really very
interesting and useful so we'll talk
about that more but for now I feel like
that was a brief introduction to
exponential equations logarithmic
equations and what their graphs look
like. Okay I'll see you in the next video.
