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We are asked to graph y is equal to log base 5 of x

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and just to remind us what this is saying,

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this is saying that y is equal to the power that I have to raise 5 to, to get to x.

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Or if i would write this logarithmic equation as an exponential equation

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5 is my base

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y is the exponent that i have to raise my base to,

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and then x is what i get when I raise 5 to the yth power.

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So anther way of writing this equation would be

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5 to the yth power is going to be equal to x.

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These are the same thing.

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Here we have y as a function of x, here we have x as a function of y.

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But they're really saying the exact same thing:

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"Raise 5 to the yth power to get x".

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But when you put this as a logarithm, you are saying:

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"To what power do I have to raise 5 to, to get x?

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Well, I have to raise it to y."

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Here, what do I get when I raise 5 to the yth power? 
I get x.

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Now that that is out of the way,

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let's make ourselves a little table

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that we can use to plot some points,

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and we can connect the dots to see what this curve looks like.

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So let me pick some Xs and some Ys.

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And in general we want to pick some numbers

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that give us some nice, round answers.

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Some nice, fairly simple numbers for us to deal with

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so we don't have to get a calculator.

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And so in general, you want to pick x values,

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you want to pick x values where the power that you have to raise 5 to to get that x value

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is a pretty straightforward power.

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Or another way to think about it is, you could just think about the different y values

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that you want to raise 5 to the power of

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and then you can get you x values.

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So we could actually think about this one to come up with our actual x values.

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But we want to be clear that when we express it like this,

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the independent variable is x and the dependent variable is y.

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We might just look at this one to pick some nice x's that give us nice, clean answers for y.

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So what happens, here I'm actually going to fill out the y first

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Just so that we get nice, clean x's.

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So let's say that we are going to raise 5 to the --I'm going to pick some new colours--

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to the negative 2 power, -- and let me do some other colours --

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negative 1, zero, 1, and I'll do one more, and then 2.

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So once again, this is a little nontraditional

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where I'm filling in the dependent variable first,

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but the way that we have written it over here, ......

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it's easy to to find out what the independent variable must be for this logarithmic function.

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So, what x gives me the y of negative 2?

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What does x have to be for y to be equal to -2?

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Well, 5 to the negative 2 power is going to be equal to x,

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so 5 to the negative 2, is 1 over 25, so we get 1/25.

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So another way, if we go back to this earlier one,

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if we say log, base 5, of 1/25.

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What power do I have to raise 5 to to get 1/25?

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Well, I have to raise it to the negative 2 power.

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Or you could say 5 to the negative 2 is equal to 1/25.

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These are saying the exact same thing.

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Now, let's do another one.

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What happens when I raise 5 to the negative 1 power?

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Well, I get one fifth. For this original one over there,

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we are just saying that log base 5 of 1/5, you want to be careful

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this is saying: "what power do I have to raise 5 to, in order to get 1/5?"

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Well, I have to raise it to the negative 1 power.

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Here, what happens when I take 5 tot the 0 power? I get 1.

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And so this relationship is saying the same thing as log, base 5, of 1,

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what power do I have to raise 5 to to get 1?

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Wel, I've just got to raise it to the 0 power.

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Let's... what happens when I raise 5 tot the first power?

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Well, I get 5.

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So if you go over here, that is just saying, what power do I have to raise 5 to to get 5?

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Well, I have to just raise it to the 1st power.

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And then finally, if I take 5 squared, I get 25.

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So if you look at it from the logarithm point of view, you say

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what power do I have to raise 5 to to get to 25?

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Well, I have to raise it to the second power.

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So, I kind of took the inverse of the logarithmic function. I wrote it as an exponential function.

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I switched the dependent and independent variables.

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So I could pick, or derive, nice clean x's that would give me nice, clean y's.

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Now, with that out of the way, but I do want you to remind,

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I could have just picked random numbers over here,

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but then I probably would have gotten less clean numbers over here, so I would have had to use a calculator.

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The only reason why I did it this way, is so I get nice clean results that I can plot by hand

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So let's actually graph it.

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So the y's go between -2 and 2,

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the x's go from 1/25 all the way to 25

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So let's graph it.

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So that is my y-axis, and this is my x-axis.

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So I'll drw it like that, that is my x-axis and then the y's

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start at zero and then you get to positive 1, positive 2,

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and then you have -1, -2 and then on the x-axis it is all positive

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and I'll let you think about whether the domain here is, well we can think about it,

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Is a logarithmic function defined for an x that is not positive?

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Is there any power that I could raise 5 to so that I would get 0?

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No. You could raise 5 to an infinitely negative power to get a very very small number

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that approaches 0, but you can never get-

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there is no power that you can raise 5 to to get 0.

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So x cannot be 0. There is no power that you can raise 5 to

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to get a negative number. So x can also not be a negative number.

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So the domain of the function right over here-

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and this is relevant becausewe want to think about what we are graphing -

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The domain here is x has to be greater than 0.

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Let me write that down.

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The domain here is that x has to be greater than 0.

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So we are only going to be able to graph this function in the positive x-axis.

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So with that out of the way, x gets as large as 25,

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so let me put those points here, so that's 5, 10, 15, 20

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and 25.

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And then let's plot these.

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... and is blue and x is 1.25 and y is -2.

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When x is 1/25, is it going to be really close to there, then y is negative 2.

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So that's going to be right over there.

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Not quite at the y-axis, 1/25 .. of the y-axis, but pretty close.

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So that right over there is 1/25 comma -2 right over there.

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Then when x is 1/5, which is slihtly further to the right,

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1/5 with y = -2. So right over there.

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This is 1/5, -1. And then when x is 1, y is 0.

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So 1 might be there, so this is the point (1,0).

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And then when x is 5, y is 1.

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I;ve covered it over here, y is 1.

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So that's the point (5,1).

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And then finally when xis 25, y is 2.

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So this is (25,2). And then I can graph the function.

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And I'll do it in the colour pink.

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So as x get's super super super small, y goes to negative infinity.

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So what power hve you have to rais e5 to to get point .0001

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That has to be a very negative power.

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So we get very negative as we approach 0,

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and then it kind of moves up like that.

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And then starts to curve to the right like that.

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And this thing right over here is going to keep going down at an ever steeper rate

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and it's never going to quite touch the y-axis.

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It is going to get closer and closer to the y-axis but it is never going to quite touch it.