Welcome, ladies and gentlemen.
So what I'd like to do is show you
how to graph exponential equations,
I'm sorry, when they're in the form of y
equals a times log base b to x, as well as y
equals a times ln of x.
And basically, these are going to have some very small trans--
well, there's actually not going to be
any-- the only transformations we're
going to have is reflections.
But I kind of want to talk a little bit as far
as the bases of the logarithm graph
as well as what the general parent graph looks
like because a lot of students get confused,
especially once we start throwing
in numbers in different places.
Even though they're not translations,
we're not shifting the graph left to right, up or down,
they are going to have some effects.
Now I don't have any graph paper.
I am just doing a general idea of what the graph is just using
the basic transformations.
So if you want to kind of see how a graph changes,
for instance, if it's log base 2 or if it's log base 4,
that is going to affect the graph
but I'm just not going to get into it.
So I do I just recommend getting into the graphing technology
you can see, but what I'm going to be dealing
with is the points on the graph that
are going to be ever changing.
It's not going to matter what the base is.
The points in the transformations
that I'm going to deal with is still
going to produce a graph that's going to be very similar, even
if those numbers are different.
For instance, same thing even if I had a number 3 here.
If it was a 3 or a 13, the graph's
going to look different, but I'm just
going to try to-- but the points that I'm going to provide
are going to be what the graph looks like so it will give you
general good idea.
All right.
But before I get into that, I would like to kind of go over
the general parent graph for you so you
can-- so we can get kind of an idea of what
the logarithmic graph looks like.
And the best way to do that is to kind of go back
to our exponential equation.
Let's just do y equals 3 to the x.
If you remember when we were graphing
the exponential equation, we always
crossed at our a, which in this case was 1.
And the graph looked something like that.
Dot, dot, dot, dot, dot.
Had of nice little asymptote, right?
Well, in my previous video, I talked
about how exponential logarithmic equations
are inverses of each other.
That means they can be reflected about the y equals x line.
Now, for the graphs that we're going to be working with today,
our a is technically always going
to be-- not our a is always going to be 1,
but the y-intercept is always going to be at 1.
So we're not going to be dealing with any functions
where it's going to be different until-- unless we
get to transformations.
So the graph of a logarithmic equation it's
going to look something like this.
OK?
So it's basically a direct reflection
across what we call the y equals x line, which
is this line right here.
All right?
So there's a couple important characteristics.
We have a x-intercept at x equals 1.
So that coordinate point is 1 comma 0.
And we also now have an asymptote.
It's a vertical asymptote.
So where we originally had a horizontal asymptote
by reflecting over the y equals x line,
we now have a vertical asymptote.
And that's very important, especially
for identifying the domain and the range for the graph.
So that's kind of like the general idea what
the graph looks like.
What I did in exponential, though,
is I created a table, which I'd like to kind of do as well.
And the main thing is for all of our graphs
until we get to transformations, this
is going to be our parent graph.
And the only thing really that changes
with-- if you have a number on the outside,
or if you have a different number in the base, that's
just going to change how the graph increases.
It can be really sharp, it can be much more shallow, and so
forth, but I'm not going to get into details
with that because I don't have a graphing calculator-- or I
mean, I don't have graph paper.
And really, the main important thing
is I want to produce a graph that's
going to be close and similar that you can
be able to identify it and understand what exactly we
did to obtain that.
Now, what I'd like to do here real quick, though,
is just kind of create a quick little table for y
equals log base 3 to the x.
And if I was going to create a table, because again,
a lot of times students get stuck
when they're trying to graph.
And the main important thing I always
want students to go back to is, remember, you can always
go back and create a table.
Now the important thing is, though,
x's-- remember the logarithmic equation says 3 raised to what
power gives you x?
Well, we want to make sure we're choosing
x values that are going to-- we want
to make sure we're choosing x values that we
can take 3 and raise it to.
So we know we can raise 3 to 3.
We know we could raise 3 to the ninth power.
We could also raise 3 to the 1/3 power.
OK?
So if I was just going to kind of sketch that, and kind of see
what would happen, here's what would happen.
So if I had 3, so if I did-- I guess
I'll erase this over here.
So I'd have y equals log base 3 of 3.
So log base 3 raised to what power gives you 3?
Well that answer is 1.
If I change that to a 9?
3 raised to what power gives you 9?
That answer's 2.
If I change this to a 1/3?
3 raised to what power gives you 1/3?
This might be a little confusing for you,
so sometimes we'll rewrite it in exponential form.
Anytime you have a fraction you can always rewrite
that as a negative exponent.
So all fractions can be written with a negati--
as a whole number with the negative power.
Therefore, using the 1 to 1 property,
which we'll get to later, we know that y equals negative 1.
So let's just kind of plot these points
and see what the graph looks like.
1, 2, 3.
Oh, I also could have taken 3, raised it to the first power,
which would have been 0.
3 raised to what power gives you 1?
That answer's 0.
All right, so we got 3 comma 1.
We have 9 comma 2.
1, 2, 3, 4, 5, 6, 7, 8, 9.
We have 1 comma 0, and we have 1/3 down negative 1.
And what you'll see is no matter how large
of-- or how big a fraction, I'm always going to get negative 1,
but it's going to keep on traveling.
It's going to keep on approaching
that y-intercept-- or the y-axis,
but it's never actually going to [INAUDIBLE],
so that's why we call-- that's going to be our asymptote.
So again, the main important thing
I want you guys to understand when we're
graphing that that I'm going to kind of take for granted, is
our y-intercepts is going to cross at 1 comma 0,
and now all I'm going to do is apply the transformations.
OK.
So.
When we-- in this case, I have y equals log base 10.
So base 10 is going to kind of affect the graph-- oh,
and also we're going to talk about domain and range.
So we know right here, my graph is going to cross at 1 comma 0.
So I'll just put a nice little point here, 1 comma 0.
And what that 3 is going to do, that's
also going to vertically stretch the graph.
But it's still just crossing at 1 comma 0.
I can't get perfect, but I just want
you to understand there's a 10 there, there's a 3,
and what that's basically doing is just vertically
kind of stretching the graph.
When we look at our domain-- now if you remember,
exponential functions-- I probably
should explain this real quick.
Exponential function looked like this.
The domain for the exponential function
was negative infinity to infinity, where
the range was 0 to infinity.
Well, notice our domain and range here
for our logarithmic equation.
The domain-- it doesn't go any negative number.
So our domain is from 0 to infinity,
whereas our range goes infinitely down
and infinitely up.
So that's from negative infinity to infinity.
And notice these are inverse functions.
Notice how their domain and ranges are basically swapped,
right?
And that's kind of the power-- that's
one of the interesting things with inverse functions,
is that their domains are the opposite--
are basically swapped.
So that's going to be really helpful a lot of times
if you want to find the domain and range-- you can't figure
out what it is for the function, you
can always go to the inverse, and find the domain and range
and then swap them to go back.
All right.
Now in this equation we have a negative.
Now if you remember, whenever we're
multiplying a negative outside of a function,
that's going to reflect over the x-axis.
When you're multiplying a function inside,
you're going to be reflecting about the y-axis.
Inside, outside, inside, outside.
Y-axis, x-axis, y-axis, x-axis.
OK?
All right.
So in this example here, we basically--
we have e, which is going to be our base, so therefore,
the graph is just going to look somewhat like this.
Oops, I should probably do that dashed, shouldn't I?
OK, it's going to look somewhere like that.
We know it's going to cross at 1 comma 0,
and it's going to have the nice asymptote.
Well, if I take this graph and reflect it about the x-axis,
graph is going to look somewhere like that.
By reflecting it, that does not change my vertical asymptote
at all, right?
I would have to be moving left or right for my asymptote
to be changed.
So the domain in this case, is again,
going to be just the same as in my original problem.
That's going to be from 0 to infinity,
and the range is going to be from negative infinity
to infinity.
OK.
A lot of times students get caught up with the base again.
And again, you can use graphing technology,
you can create another table, and you
can see that this graph is going to have different slopes
than this graph.
Multiplying by 3 on the outside is going to affect the parent
graph as well as having a base 2 is going
to affect the parent graph.
However, for the problems that we're working on, all of them
are still going to cross at 1 comma 0.
So to kind of understand how those truly [INAUDIBLE]?
My best thing, what I do in my class,
is just use graphing technology so you can
see those kind of differences.
But they're not going to be dramatically changing
the graph.
What we're focused on for the changes of the graph
is going to be our negatives.
However, this 3x, though, does-- will affect it,
which we'll talk about.
OK.
OK, that works.
Well, OK.
So in this case, now let's go ahead and graph.
Again, there's really nothing that's affecting.
My y-intercept is still going to be 2.
And again, if you don't understand
why the y-inter-- why the x-intercept's always 2,
again, plug 0 in for y, right?
Here.
Here is the x-axis, here's the y-axis.
We want to know what the y-intercept is.
So negative log base 2 of x.
Well, divide by negative 1.
0 equals log base 2 of x.
2 raises it.
What to the 0 power equals x?
Well, x equals 1.
OK?
The y-- the y asymptote, or the vertical asymptote,
is still going to be the same.
Again, we're multiplying by a negative on the outside.
So that's going to reflect the x-axis.
So it's going to look something like this.
Nice little vertical asymptote.
Where now we can say the domain is
going to be from 0 to infinity, where
the range is going to be from negative infinity to infinity.
All right.
Looking over here, I have y equals 1/2 ln of x.
So whereas the 3 would kind of vertically stretch the graph,
the 1/2 would kind of compress it vertically.
So therefore, again, using the same kind of process and idea,
I know that this graph, even though it has a base e,
is still going to cross-- is still going to cross at 1.
And again, I can prove that for every single problem
just by plugging in 0 in for y and solving.
So it's going to look something like this.
And the 1/2 is just going to kind of stretch--
or vertically compress it a little bit.
But we still have my asymptote here, I can still
say my domain is from 0 to infinity,
and my range is from negative infinity to infinity.
OK, so now we're going to have some changes in our domain
and range, because now I'm multiplying
by an x inside the function, which
is going to be reflecting about the y-axis, not just the x.
So again, first thing I like to do
is just graph the-- graph the function
without any transformations.
So in this case, again, by rewriting this,
I know that I am going to have 1 as my y-intercept.
OK?
But if I was going to take this graph
and flip it about the y-asymptote, or y-intercept,
the graph would look something like this.
However, even by reflecting it, my y-- my asymptote
does not change, right?
You're reflecting it about the y-axis, which
is where your asymptote is, so that's not going to move.
That's not going to change.
So my domain for this problem-- now, the graph
is going infinitely to the left.
So it's going to be from negative infinity,
and it's going to go all the way to the right,
but it's not going to cross my asymptote.
So it's going to be negative infinity to 0, whereas my range
is still going to be the same.
They're both going infinitely down
and they're both increasing infinitely up.
So it's going to be negative infinity to infinity.
OK, now the last example.
This one can be a little tricky because I basically told you
for every single one of these problems,
your x-intercept is always equal to 1.
And that really is the case.
I just didn't show the work for each one
because it's redundant.
However, for logarithms, we got to be a little bit careful.
When we have a value that's being multiplied inside
of a function, that is going to affect our x-intercept
and I'll show you why.
Again, do the same thing that I did up here.
Replace y in for 0 equals ln of 3 to the x.
Rewrite it in exponential form.
e to the 0 equals 3 to the x.
Divide by 3.
1/3 equals x.
So now my x-intercept is actually 1/3, not 1.
But I have no other reflections or no other changes.
So if here's 1, 1/3's going to be like that.
It's still going to be approaching 0 as its asymptote,
and still going to be infinitely to the right
and infinitely down, so my domain
is-- domain and range are not affected.
So again, as I remember, if you get confused or kind of forget,
always go back to creating a table.
Hopefully you also have graphing technology,
which you can always double check your answers
and so forth.
But this is just kind of a basic idea
of how to determine domain and range,
as well as how to graph your logarithmic equations
with different bases, coefficients, as well
as reflections.
Thanks.