BAM!!! Mr. Tarrou. In this lesson we are going
to do three examples of graphing logarithmic
functions using tables. We are not going to
be allowed to use our calculator. A couple
of years ago, two or three years ago I did
a video related to this same topic. I spent
a lot of time explaining the inverse relationship
between exponential functions and logarithms
functions. I will have a link to that lesson
in the description. I finished off with an
example very similar to to how we are going
to work these problems. But then the last
one or two examples I did was just sketching
these functions off of transformations. You
know, knowing the parent function and moving
it left and right, up and down, flipping it,
and whatever. Well if you don't know what
the parent function looks like, maybe you
know what y equals log base two looks like,
but what happens if the base is equal to one
half? What happens if there is a leading coefficient
or a plus at the end? Or maybe there are so
many transformations involved in the equation
that your sketch is uh...of these functions
just based on transformations and knowing
what the log, or excuse me...the parent function
looks like, just ends up being a little bit
rough. I don't want a rough sketch, I want
a really nice graph. At least know five points
that these go through and find those five
points without the aid of a calculator and
again get that really nice accurate graph,
not a rough sketch just based off of transformations.
As we finish these three examples we will
talk about the domain and range of these graphs
as well. We are going to put these first two
graphs on the same x y coordinate system because
I just want to point out the differences and
effect of when we have a base that is larger
than one and what your logarithmic graphs
look like when your logarithm has a base that
is between 0 and 1. We also have a vertical
stretch which is an a value which is equal
to three. But, I really want to talk about
what happens with these different bases. Remember
that you cannot take the log of a negative
number. I am just reviewing some logarithmic
facts for you. The answer that you get from
a logarithm is an exponent. Hopefully if you
are watching this video you already know that.
I like to mention that the common log, log
base ten of a hundred is equal to two because
the base of that log is 10, common log, ten
squared is a hundred. I am pointing in here
like it is base ten. If I do log base ten
of 100, we get an answer of 2. That is an
exponent. It is ten squared gets you back
to a hundred. There is at least a five second
review of how logarithms are related to exponents.
Ok, so when you look at logarithms...when
you convert from log form to exponential form
you need to remember of course that the base
of the logarithm is going to be the base in
exponential form. You do get exponents from
logarithms. So we will have say two to the
y, and then the x plus one is an answer, not
as far as far as having this written in logarithmic
form but it will be the answer when you get
this written in exponential form. That is
exactly how we are going to graph these three
examples without the aid of a calculator.
We are going to convert them into exponential
form, build up a t-table, pick out some smart
numbers so that we can do it in our head,
and plot those points and get our graph. Now,
just one more time. If you do know your parent
functions and you know your transformations,
then you will know the difference between
y equals log base 2 of x plus one and y equals
log base two of just x. It is a horizontal
shift to the left or a horizontal translation
to the left of one unit. So let's see what
this looks like though doing this problem
with a t-table. We have a base, we have an
exponent, and we have an answer. That is again
how we are going to do these problems without
the aid of a calculator. So we have two to
the y power is equal to x plus one. We see
that y variable is up here in the exponent
and is it is sort of where the math is happening.
We are going to therefore... We are not going
to re-isolate the variable y, it already was
in the first place. I guess you could make
it a log base ten or log base e, but again
I want to do this without the aid of a calculator.
I can take the base of two and raise it to
certain powers in my head. So we are going
to solve for x, and we are going to do that
by subtracting both sides by one. AND we get
two to the y minus one is equal to x. Now
I said that we are going to do this with tables.
Normally when you do a t-table you pick the
x values, but that is normally the y is along
and the x is over there where the math is
happening. Well now the x is alone and it
is the y that is involved with all the math.
Two to the y power, now there is a minus one
with it. So when we build up our t-table we
are going to be picking values of y. Now what
values do you want to use? Pick some small
numbers around the zero range and use integers
more than likely. We want to pick some numbers
that can easily raise two to that power. Let's
just say that y is going to be negative two,
negative one, zero, one, and two. I don't
usually just use two or three points for graphing
a function that is curving. Right, what direction
is it curving, how sharply is it doing that?
So let's make up some...let's do some work
here. So y is equal to negative two. That
means that we are going to have two, the base
of two to the exponent of negative two minus
one is equal to x. Two to the negative two
power. How do you deal with, or get rid of
a negative exponent? How do you rewrite this,
or move this base around so this exponent
is positive two as opposed to negative two?
Well you move the base. If the base is in
the numerator with the negative exponent,
you drop it down. If the base is in the denominator
with a negative exponent, you will move it
up to the top of the fraction. And if you
understand division and the properties of
exponents, you can hopefully understand how
moving the base up and down will change the
sign of the exponent. But, this will be one
over two squared minus one. That is going
to be two squared of course is four. Now we
are looking at 1/4 minus one. When you start
dealing with fractions you should make everything
look like a fraction. So now we need a common
denominator. We are going to multiply the
numerator and denominator of the one by four.
We have now one over four minus four over
four is going to be equal to negative three
fourths. When we have a y value of negative
two our x value is -3/4. When we let our y
value be negative one, we are going to have,
for this next line, we are going to have two
to the negative one. We are going to have
to rewrite this to get rid of the negative
exponent, as one over 2 to the first, or just
one over two minus one over one. Now we are
going to combine these fractions. Multiply
the numerator and denominator by two giving
us 1/2 minus 2/2. When you have common denominators,
you just add or subtract those numerators.
One minus two is equal to negative one....half.
Now we are going to let y equal zero. And
anything to the zero power is going to be
equal to one. We are looking at one minus
one which is going to be equal to zero. Then
we are going to let y equal one. And this
is going to go a lot quicker because we are
not dealing with fractions any more. Two to
the first is two minus one is one. And then
when we let y equal two... I guess I could
have rewritten that line as I was going...
Two squared is four and four minus one is
three. That was pretty easy. I don't need
a...require a calculator to do that. And...
I can't do base of two!!!:( My calculator
only does base e and base ten. Who cares.
You don't even need a calculator. So let's
see here. X is going to be -3/4. Each of the
tick marks count for one. X is equal to -3/4
and y is going to be -2, so I am going to
try my best to estimate that. We have an x
values of -1/2 and a y value of -1. We are
passing through the point of (0,0). Logarithmic
functions, if there has not been a horizontal
or vertical transformation, we have a plus
or minus here inside the math function to
move the graph to the left, if there has been
no horizontal or vertical translation or slide,
log functions will always pass through the
point of (1,0). Then we have (1,1) and (3,2).
Now, by definition logarithms must have a
base which is positive. If they had a base
which was negative, it is nice to talk about
it now that we have this in exponential form.
A negative number raised to a power, again
we are talking about functions right...some
kind of graph that would look like a function
and pass a vertical line test... If you have
a base which is negative and you raise it
to an even power, the answer is going to be
positive. Forget the plus or minus something
constant going on. Then when you take that
negative base, that base of the logarithm
is the base of your exponent, a negative number
raised to a...what did I just say? An even
number is positive and an odd power is negative.
So if you had that negative base, or a negative
base in an exponential function the graph
would be oscillating back and forth violently
between positive and negative values and you
wouldn't get a smooth curve or function. So
that is why your bases have to be positive.
If the base of your logarithm has to be positive
and what you get out of it is an exponent,
so two to something has to be equal to what
is inside that parenthesis, you are not going
to take a positive number raise it to any
power... I just took, I erased it, but I took
a positive base of two and raised it to the
negative two power, I didn't get a negative
answer. That is negative but that was after
I subtracted by one. You are never going to
take a positive base raise it to any power
and get a negative answer. So you cannot log
zero and you cannot log a negative number.
So logarithms are always going to have a vertical
asymptote as part of their graph. We can see
that the graph is curving down and the first
one which I really intended on doing in yellow
is going to look like this. Instead of the
vertical asymptote being at x equals zero
it is going to be at x is equal to negative
one because of that horizontal shift to the
left. I didn't even have to think about that!
The table of values allowed us to see that
developing and our graph looks something like
this. Now, it is sort of a sketch and sort
of a graph. I mean in this part of the graph
it is really accurate but now...you know..I
am not sure how quickly it is going to climb
as we go to the right. But it is going to
look something like that. So now we have got
our first graph done. Y is equal to log base
two of x plus one. Now, you might have thought,
"I know what the parent function looked like.
I knew it was going to move to the left anyway."
But, that is not just a sketch, it is more
of an accurate graph. Do you know the parent
function for a base of one-half, and with
that vertical stretch of three. So, let's
do this with a t-table. How is that base of
1/2, the base being between 0 and 1, going
to compare to when we had a base which is
bigger than one. Let's just put into exponential
form. We are going to... You cannot put this
into exponential form until you get the log
function alone. I want to have a leading coefficient
of one, and then convert it into exponential
form. I could bring the three up as a power
and say that it is y equal log base 1/2 x
to the third and then convert it into exponential
form. But my y is going to have some kind
of exponent and my x is going to have some
kind of exponent, and I want to just plug
some values in for y and get x. You know,
like get the answer for x. I just want the
x alone. So we are going to divide both sides
by 3. I don't know why I picked up green when
the problem is in purple. But we have y over
three is equal to log base one half x. Now
we have x is nice and alone. That is going
to be handy for when we make our t-table.
The leading coefficient of the logarithm is
equal to one. So now we are going into exponential
form which is base, exponent, answer. So we
have one half raised to the y over three power
is equal to x. Now you think, well that looks
a little bit nasty but you have control when
you are building a t-table of what you are
decide to set for your values of y. Your y
is your independent variable and x is dependent
variable, at least the way the equation is
structured. So when we set up or t-table,
and I want to do this without the aid of a
calculator why not just pick values of y that
you can divide by three like 6, 3, and 0.
So we have x and y and y is going to be negative
six, negative three, zero, three, and six.
Now let's plug these in and find out what
happens. Let's see. Let's get the negatives
out of the way first. We are going to have
negative six, so negative six divided by three
is going to be a power of one half. That is
going to be one half to the negative two.
If you want to rewrite your base so that it
does not have a negative exponent you flip
that base. We are going to have two to the
positive two power. Two squared is equal to
four. Repeat the process with negative three
and we are going to have negative three. Negative
three divided by three, that is coming in
from here right, is equal to negative one
and that is going to be equal to two over
one to the first which is of course just equal
to two. Now, when we put in that y value of
zero, we are going to have... Letting y equal
zero we have 1/2 to the zero over three power.
Zero can be on top, it just cannot be in the
denominator. Zero divided by three is zero.
We have one half to the zero power which is
anything to the zero power is one. That is
true because...let's see here. How about,
what is x to the third...this is just an example
of...what is x the third divided by x to the
third? We know that anything divided by itself
is equal to one right? Well when you divide
like bases and if I didn't want to just say
...ohh anything divided by itself is one...When
you divide like bases, don't you subtract
the exponents? Wouldn't that be x to the zero
power because three minus three is zero? Yeah
anything to the zero power is is one. I don't
if you knew that or not. Some of my teachers
in high school used to just always say, "It
is just a rule, memorize it." We have, opps
I need to put the answer there. Now we are
going to let y equal three. Look at that,
I already have the bottom of the three in
there. We are going to let y equal three.
Three divided by three is equal to one and
we have one half to the one power which of
course is one half. When we let y equal six
we are going to have six divided by three
which is two. One half squared is one half
times one half. I don't think you really need
to see that expanded, but in case you do one
times one is one and two times two is four.
Alright, so let's see here. When x is equal
to, we are going to do this in purple for
the graph, when x is equal to one y is zero.
I don't know why I am doing that point first.
When x is equal to 1/2 y is equal to three.
So when x is 1/2 y is equal to three. See
how the graph is not rising up to the right?
It seems to be falling. When x is equal to
1/4 the y coordinate is all the way up to
6. When x is two, the y value is negative
three. And when our x value is four the y
coordinate is negative six. So x is four,
and y is negative six. That graph is going
to look something like. More than something
like this, it is actually going through these
points, but I don't know how quickly it bends
over as it goes to the right. To answer that
I would need to make a bigger t-table. But,
hmm... Ooo, I said I was going to talk about
domain and range. Now notice that the base
being between zero and one, this graph is
falling. Let me check the time by the way.
Ok, so with that base bigger than one, and
most of the time your bases are going to be
bigger than one, those logarithms come up
from negative infinity and go over to the
right. With that base of one half, that is
going to get a function that is monotonic
decreasing as opposed to monotonic increasing.
It is interesting. Now the domain and range
of this yellow graph. The domain is going
to be, well it goes all the way to the left
and it approaches negative one, but x equals
negative one is a vertical asymptote. The
x's that make up this graph, this yellow graph,
are never going to become negative one but
it will approach negative one. So the domain
are your x's between negative one, oops almost
wrote infinity, negative one to infinity.
The graph as you can see goes down and it
is sloping over sharply to the right but it
will never stop increasing. So this yellow
graph is going to go up and down forever,
so the range which are the y values are going
to go from negative infinity to positive infinity...
AND BEYOND! The purple graph, well it goes
up and down forever as well so it's range
is going to be the same. It's domain though,
we have a vertical asymptote at x equals zero
so this purple graph is going to have a domain
of zero to infinity. Now I just have one more
example and really it is same kind of deal,
just a more complicated example. So I am going
to step off, put that question up here and
just reveal the solution one step at a time
so that you could pause the video, try it
on your own, and see if you can't get that
logarithmic equation written into exponential
form and check your work. Because I am going
to just reveal the next example in small steps
at a time, because really the process is the
same. Be right back, BAM!!! Nanananana...
So we have our last example. It was y is equal
to negative two plus log base three of x plus
four. As you saw, trying to isolate that logarithm
we added both sides by two. Then we went into
exponential form, base-exponent-answer. And
then we had to subtract both sides by four
until we completely isolated the x. Ok, I
cheated a little bit and used a calculator
to figure out that 35 over 9 was 3.9 so I
did not have to do the long division by hand
and so on for the remaining decimal. I did
really intentionally want my x values to be
so close together and be at least a tiny bit
difficult to graph. I just looked at the exponent
of y plus two and thought, well why don't
I pick my y values so that the middle term
is going to have an exponent of zero, and
we will have a couple of negative exponents
and a couple of positive exponents. That is
how I came up with the idea of using these
y values for my t-table. Also, i went ahead
and before putting the vertical asymptote
on our graph, what is inside the logarithm
again has to be positive. It has to be greater
than zero...greater than and not equal to.
x plus four is greater than zero, subtract
both sides by four and we get the placement
of our vertical asymptote, the domain, put
the points down, and that is the end of our
last example. I hope this helps. I am Mr.
Tarrou. BAM!!! Go do your homework:D AND HAPPY
PI DAY!!!
