We're asked to graph f(x) is equal to the natural log of
2x, and I'll use a calculator here to find a table of values
for different x values, what does f(x) equals, but then I'll hand draw the graph
which might be a little bit ironic
because I'll be using a graphic calculator to come up with the values.
But I won't use the graphing function to graph it,
we'll do that part by hand.
So let's draw here a little table of values.
Of x values and of y values.
And see what we get.
So these are my x values and let's just say y = f(x)
So whatever the output is I'm going to set that equal
to our dependent variable and plot it on the vertical axis
and we call that dependent variable y.
So let's just try really small numbers.
So actually first, let's remind ourselves what's the domain?
What's the set of valid inputs for x, that we can put in right there?
So the natural logarithm is just logaritm base e.
And any logarithm is only defined when input into the logaritm,
in this case 2x, is greater than zero.
You can't even take the logarithm of zero.
You can raise something to a very negative exponent,
negative bilion power, it will get pretty close to zero
but you can never get to zero.
If you take a positive base there is no exponent you can raise it to
to get to the zero or get to a negative number.
So the 2x here, the input into our logarithm function,
in this case the natural logarithm function,
it has to be greater than zero.
And if that's greater than zero, divide both sides by 2.
That means x is greater than zero.
So that is essentially the constraint on our domain.
Our domain is all real numbers greater than zero.
So let's try something pretty close to zero.
Just so we see what happens, the behaviour as we approach
as we are close to zero.
And especially as input here is less than one.
So let's try 0.1, 0.5, 1.
And let's try, I don't know - let's try 5...
Oh, I don't actually want to get too far,
cause I want to be able to see the resolution down here.
So let's try 1.5 and let's try 3.
So those will be our inputs that we'll try.
That's sounds good enough.
And then let me draw my axes and then we'll plot the points.
So our domain is positive x values
so we don't really have to draw much on the negative x values.
But we will have some negative values here,
so let me give ourselves some room to work with.
And out x values go up to 3.
So this is 1, 2 and 3.
This is 0.5, 1.5 and this is 2.5 (which we do not use).
And then let's see what our y values are, f(x) are going to be equal to
So get out our TI-85
So if we take the natural log -- remember, we have to take
2 times x and then natural log of that.
So if we take the natural log of 2 times 0.1,
which is obviously 0.2, what do we get?
We get negative 1.61, I'll say.
Negative 1.61.
And then if we input 0.5, what do we get?
Let me get the calculator back.
So we're going to take the natural log of 2 times 0.5.
We can do it in out heads.
Actually I'll just write it out just so it's clear what we're doing.
2 times 0.5, that's the natural log of one.
And you should be able to do that in your head.
What power you have to raise any positive base to, to get to one?
Well, you raise e to the zero power and you get one.
So I'll write it over here.
We should've been able do to that one in our heads.
Now let's do the next one.
What happens when we have the natural log of 2 times 1.
Which is obviously just going to be 2.
So it's te natural log of 2.
Gets us .69. Which makes sense that this is less than 1.
Because 2 is less than e. e is 2.71 and so on and so forth.
So this is .69.
Now let's try the natural log of 2 times 1.5.
Natural log of 3.
And that gets us to 1.10, I'll round to the hundedths.
So this is 1.10.
And finally the natural log of 2 times 3.
2 times x, x is 3.
What do I get? This is going to be natural log of 6.
Which is 1.79.
This is going to be, I'll choose a new color.
This is going to be 1.79.
So in terms of the coordinates we run as low as -1.61 and as high as 1.79.
So let's call this right over here -1
and then down here would be -2.
I'll extend the y axis down a little bit.
So this is the x axis and this is out y = f(x) axis.
And then let's call this right over here positive 1.
And this over here is positive 2.
And this would be half way between those
just cause it looks like we gonna have to be able to see that as well.
And so this first point is 0.1, -1.61.
So 0.1, that's one tenth,
that's gonna be right around there.
And then we have -1.61.
It's gonna sit right about there.
So that is the point (0.1, -1.61). Fair enough.
That's the first point right there.
Now let's do this one.
0.5, 0
When x is 0.5, y is 0.
That's (0.5, 0). Fair enough.
When x is equal to 1, y is 0.69.
Which might be right about there.
Just approximating it, so a little bit closer to 1 that is close 0.5.
Actually a little bit closer to 0.5 than it is to 1.
So let me put it right over there.
This would be (1, 0.69)
And then we have the point, when x is 1.5 - f(x) is 1.10.
When x is 1.5, f(x) is 1.1, which takes us right above there.
So that is the point and I think you see where this curve is going.
(1.5, 1.10)
And finally, so that was that point.
We'll do this last one in yellow
when x is 3, y is 1.79.
So a little bit closer to 2.0 than to 1.5.
So it's gonna be right about there.
It's gonna be coordinate (3, 1.79).
And now we can connect the dots
and I'll do that in white.
And so as we get x values that are closer and closer to 0
Our graph of our function is gonna get more and more negative
And it's gonna get closer and closer to the y axis without ever touching it.
So it's gonna get less close from the y axis, slowly break away
and then curve out like this.
Curve out just like that.
And just keep on going down like this.
And what happens here is pretty cool, as x gets smaller and smaller
The functions becomes more...infinetely negative as x aproaches 0.
But x can never be 0.
There is no power you can raise e or any positive base to,
to actually get zero. You can raise it to raise to a very large negative exponent
I you raise e to the negative one billion, you'll get a number
That's very close to 0, because it is the same as
1 over e to the one bilionth power.
So this is a number that's very close to 0.
But you're never going to approach zero.
You can make this number more and more negative
it'll just get smaller and smaller numbers
but you'll never quite approach 0.
So you can never a logarithm of zero. You just approach it.
We're done! This is the graph of natural log of 2x.
And it has a typical shape, cause it's just a logarithm.
It has a shape of a logarithmic graph.