Graphing natural logarithm function Graphing Natural Logarithm Function
Graphing natural logarithm function
- 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.
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