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We're asked to graph f(x) is equal to the natural log of

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2x, and I'll use a calculator here to find a table of values

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for different x values, what does f(x) equals, but then I'll hand draw the graph

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which might be a little bit ironic

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because I'll be using a graphic calculator to come up with the values.

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But I won't use the graphing function to graph it,

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we'll do that part by hand.

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So let's draw here a little table of values.

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Of x values and of y values.

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And see what we get.

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So these are my x values and let's just say y = f(x)

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So whatever the output is I'm going to set that equal

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to our dependent variable and plot it on the vertical axis

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and we call that dependent variable y.

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So let's just try really small numbers.

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So actually first, let's remind ourselves what's the domain?

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What's the set of valid inputs for x, that we can put in right there?

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So the natural logarithm is just logaritm base e.

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And any logarithm is only defined when input into the logaritm,

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in this case 2x, is greater than zero.

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You can't even take the logarithm of zero.

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You can raise something to a very negative exponent,

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negative bilion power, it will get pretty close to zero

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but you can never get to zero.

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If you take a positive base there is no exponent you can raise it to

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to get to the zero or get to a negative number.

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So the 2x here, the input into our logarithm function,

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in this case the natural logarithm function,

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it has to be greater than zero.

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And if that's greater than zero, divide both sides by 2.

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That means x is greater than zero.

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So that is essentially the constraint on our domain.

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Our domain is all real numbers greater than zero.

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So let's try something pretty close to zero.

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Just so we see what happens, the behaviour as we approach

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as we are close to zero.

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And especially as input here is less than one.

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So let's try 0.1, 0.5, 1.

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And let's try, I don't know - let's try 5...

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Oh, I don't actually want to get too far,

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cause I want to be able to see the resolution down here.

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So let's try 1.5 and let's try 3.

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So those will be our inputs that we'll try.

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That's sounds good enough.

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And then let me draw my axes and then we'll plot the points.

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So our domain is positive x values

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so we don't really have to draw much on the negative x values.

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But we will have some negative values here,

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so let me give ourselves some room to work with.

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And out x values go up to 3.

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So this is 1, 2 and 3.

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This is 0.5, 1.5 and this is 2.5 (which we do not use).

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And then let's see what our y values are, f(x) are going to be equal to

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So get out our TI-85

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So if we take the natural log -- remember, we have to take

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2 times x and then natural log of that.

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So if we take the natural log of 2 times 0.1,

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which is obviously 0.2, what do we get?

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We get negative 1.61, I'll say.

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Negative 1.61.

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And then if we input 0.5, what do we get?

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Let me get the calculator back.

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So we're going to take the natural log of 2 times 0.5.

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We can do it in out heads.

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Actually I'll just write it out just so it's clear what we're doing.

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2 times 0.5, that's the natural log of one.

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And you should be able to do that in your head.

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What power you have to raise any positive base to, to get to one?

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Well, you raise e to the zero power and you get one.

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So I'll write it over here.

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We should've been able do to that one in our heads.

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Now let's do the next one.

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What happens when we have the natural log of 2 times 1.

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Which is obviously just going to be 2.

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So it's te natural log of 2.

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Gets us .69. Which makes sense that this is less than 1.

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Because 2 is less than e. e is 2.71 and so on and so forth.

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So this is .69.

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Now let's try the natural log of 2 times 1.5.

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Natural log of 3.

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And that gets us to 1.10, I'll round to the hundedths.

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So this is 1.10.

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And finally the natural log of 2 times 3.

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2 times x, x is 3.

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What do I get? This is going to be natural log of 6.

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Which is 1.79.

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This is going to be, I'll choose a new color.

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This is going to be 1.79.

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So in terms of the coordinates we run as low as -1.61 and as high as 1.79.

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So let's call this right over here -1

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and then down here would be -2.

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I'll extend the y axis down a little bit.

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So this is the x axis and this is out y = f(x) axis.

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And then let's call this right over here positive 1.

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And this over here is positive 2.

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And this would be half way between those

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just cause it looks like we gonna have to be able to see that as well.

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And so this first point is 0.1, -1.61.

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So 0.1, that's one tenth,

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that's gonna be right around there.

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And then we have -1.61.

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It's gonna sit right about there.

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So that is the point (0.1, -1.61). Fair enough.

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That's the first point right there.

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Now let's do this one.

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0.5, 0

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When x is 0.5, y is 0.

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That's (0.5, 0). Fair enough.

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When x is equal to 1, y is 0.69.

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Which might be right about there.

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Just approximating it, so a little bit closer to 1 that is close 0.5.

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Actually a little bit closer to 0.5 than it is to 1.

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So let me put it right over there.

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This would be (1, 0.69)

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And then we have the point, when x is 1.5 - f(x) is 1.10.

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When x is 1.5, f(x) is 1.1, which takes us right above there.

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So that is the point and I think you see where this curve is going.

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(1.5, 1.10)

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And finally, so that was that point.

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We'll do this last one in yellow

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when x is 3, y is 1.79.

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So a little bit closer to 2.0 than to 1.5.

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So it's gonna be right about there.

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It's gonna be coordinate (3, 1.79).

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And now we can connect the dots

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and I'll do that in white.

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And so as we get x values that are closer and closer to 0

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Our graph of our function is gonna get more and more negative

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And it's gonna get closer and closer to the y axis without ever touching it.

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So it's gonna get less close from the y axis, slowly break away

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and then curve out like this.

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Curve out just like that.

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And just keep on going down like this.

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And what happens here is pretty cool, as x gets smaller and smaller

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The functions becomes more...infinetely negative as x aproaches 0.

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But x can never be 0.

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There is no power you can raise e or any positive base to,

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to actually get zero. You can raise it to raise to a very large negative exponent

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I you raise e to the negative one billion, you'll get a number

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That's very close to 0, because it is the same as

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1 over e to the one bilionth power.

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So this is a number that's very close to 0.

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But you're never going to approach zero.

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You can make this number more and more negative

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it'll just get smaller and smaller numbers

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but you'll never quite approach 0.

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So you can never a logarithm of zero. You just approach it.

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We're done! This is the graph of natural log of 2x.

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And it has a typical shape, cause it's just a logarithm.

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It has a shape of a logarithmic graph.