HELLO, Mr. Tarrou. We are graph some logarithms but first we are going to graph an exponential
function. I did this exact same one in a video just about 20 minutes ago. I don't know when
you watched or even if you did..haha. Anyways. Y equals 2 to the x power. Ok So I have picked
out some x values that I want to plug into this function. The reason I am picking my
x values is because the x is where the math is happening and it is 2 to the x power. So
that is the easiest thing to figure out. Two to the negative one is one half because when
you have a negative one power it will move that base to the denominator. Two to the zero
power is one. Two to the first is two. Two the the second is four. Two to the third is
equal to eight. So, we have (-1,1/2). Oops, I forgot my tick marks along here, hold on
a second. Thought I was ready! Ok, negative one and one half. (0,1) We have got (1,2)
And we have got (2,4) and (3,8) One, two, three, four, five, six, seven, and eight.
Ok. And talked about how the exponential growth curve, if they are in the basic form of y
equals a to the x power, that they have a horizontal asymptote of y equals zero, they
all pass through the point of (0,1), and because my base is larger than one it is growth. Awesome!
You heard me talk a little bit, in the last video if you watched it, about intro to logarithms
that logarithms and exponential functions are sort of inverses of each other. Log of
a 100 is 2 and... Let me clarify that. Log base ten of a hundred is two and ten squared
will undo that log function and get you back to 100. So, they are inverses functions of
each other. How do you find inverse functions graphically. Like if I gave you this graph
and said make the inverse function, what would you do? Well hopefully you would remember
that you would take the x and y values and flip them. So we are going to do that graphically
and through the algebra to let you see this really develop and remind you that if you
want to find an inverse function you flip x and y variables. I am going to take the
x and y's and go x equals two to the y. Now if I actually wanted to graph the inverse
function, not graph it.. I am going to graph it... If I wanted to actually find the inverse
function I would need to solve this for y. I don't want to bother with that, I just want
to graph the inverse function. Ok.. There was something I was going to say and I forgot.
Well, if I am going to do a t table to make this graph, what would I want to pick? Would
I want to chose some x values and then try to solve with those for y. If I pick the wrong
x value it is going to take me a logarithm to solve for the value of y. We have not gotten
to properties of logarithms and solving equations using logarithms yet. We are still trying
to get used to logs. I definitely do not want to make this difficult. What I am going to
do is pick my values of for y because again the y is where the math is happening. If I
just want to make a graph, I don't want to pick an x and solve for y backwards all the
time. No, lets just... I switched the x and y's around, why not just pick the y's instead
of the picking the x's like we did before. You might notice that I am using the same
numbers again. (there is a reason for that) So two to the negative one is one half. Two
to the zero is one. Two to the first is two. Two to the fourth is four. Two to the third
is eight. So I am the same thing I said a second ago except what was my y values are
now my x's and vise versa. Well that is what you do when you graph an inverse function.
Now I remember what I wanted to say, and it is not this but... When you graph and inverse
it is a reflection over the line y equals x. The reason... Here is what I wanted to
talk about. This white graph, this y equals two to the x, this exponential growth function,
this will have a inverse FUNCTION because not only is it a function itself, it will
pass the vertical line test, but it will also pass the horizontal line test. This is a one
to one function. I talked about that concept when we did that earlier this year as well.
If you don't know about one to one functions you can go to my little search window in the
upper right hand corner and will see a video about that. I want to reflect this now over
the line y=x. I want to graph the inverse function. So, (1/2,-1)... oops... yellow...
(1,0) (2,1) (4,2) and (8,3) Ok. There is my graph. Whooptie Doo! What used to be my horizontal
asymptote of y=0 has now become my vertical asymptote of x=0. And you might start thinking
to your self, hey Mr. Tarrou you made a graph but the title says graphing logarithms and
I don't see any logarithms up here. Well, yes you do! While this white line is exponential
growth, this is the inverse of the exponential growth function. I say that this is a log..
a graph of a log function. The log function is the inverse of the exponential function.
I am going to prove that to you by going to this equation and writing this equation that
is in exponential form and we are going to write that in log form. So log...Maybe I should
use the same color. Log, what is the base of the exponent?...2, so log base two. Log
base...in the exponential form you actually logging sort of the answer... so the answer
was x. What did I chant in the last video? What do you get out of a logarithm? An exponent.
What do you get out of a logarithm? An exponent. Why is the log of a 100 equal to 2? Because
it is 10 squared which equals 100. So you get an exponent out of a logarithm and voila.
Log base two of x is this graph. So in textbook if you are asked to graph a log function,
you are going to want to actually not graph it as a logarithm because you are probably
are going to have a base that your calculator cannot work with. Now if it is y equals natural
log or y equals common log something x then you can put that in your calculator. But if
it is a base other than ten or e you are probably going to have to into exponential form and
then make a t table and graph it. You will pick the y values and not the x. So I have
kind of already graphed your log function without telling you I was. Pretty Cool!!!
So that yellow graph is y equals log base two of x. So what if I wanted to graph y equals...
Let's pick a different color. How about y equals log base two of x minus 3. What would
that look like? Well a plus or minus inside your math function that is a horizontal movement
left or right. This is a minus three so that is a translation to the right. So I can use
this parent function to graph this function. Y equals log base two of x minus three is
going to take all these points and move them over three places. This one... one, two, three.
This one... one, two, three. This point over three units. This point over three units.
This point over three units, it is in the middle so it will be over here somewhere.
So we have that. Good enough! If you were starting from scratch, and you did not know
what the parent function looked like you could still write this in exponential form. This
is two to the y power is equal to x minus three. Write that as two to the y power plus
three equals y. And then still do your t table and pick out your own y values. Ok... So there
is a couple of ways to graph that. Let me erase all this stuff and talk about some properties
of these graphs for you before we run out of time. We will just erase all of this. That
was y equals log base...that could be anything even natural but I am going to keep the log
base two we had up there a second ago. Actually no I am not, I am just going to put that as
any base. Y equals log base b of x. It has... All these graphs will pass through the point
(1,0) Not (0,1) like the exponential growth and decay ones, but (1,0) because that reflection
over the line y = x. They have instead of a horizontal asymptote, they have a vertical
asymptote of x=0. Let's talk about that for a second. Let me finish this first. They all
pass through (1,0), they all have a vertical asymptote of x=0 unless there has been a horizontal
shift. They have a domain that is from zero to infinity. Because the vertical asymptote
the graph starts just to the right of zero and goes to the right forever. And the range
that is all real numbers. Negative infinity to positive infinity. ehh... goes up and down
forever so all the y values are used. Again unless there is a horizontal shift, why can
you not log a negative number? Why is there no graph on the negative side of the x axis?
Let's see. Let's go back to that original equation that I had up there of y equals log
base two of x. Let's write that in exponential form two to the y is equal to x. How can there
be no negative x values? Because the base is positive!!! Y is an exponent of that positive
base. Let's say it is two to the second, two times two. Positive time positive. Let's say
it was two to the third. 2, 4, 8 See, the exponent is tell you how many to multiply
by that positive base. You are never going to repeatedly multiply by a positive number
and get a negative answer. So thus, you cannot log a, because logarithms must have a positive
base by definition, so you cannot take the log of a negative number. Because, that positive
base of the log can be raised to no power and ever become negative. Even if it was...
Let's say y is negative one have.... that will be too hard. How about negative two.
Two to the negative two. That does not come out to be negative. It would become one over
two squared which is one fourth. And again you it would be positive. So when you have
a positive base you can raise that to no power what so ever that would ever give you a negative
answer. So the standard domain, and again there can be a horizontal translation, but
the standard domain of log functions start to the right of zero and go forever. I am
Mr. Tarrou. BAM, GO DO YOUR HOMEWORK!!!