Graphing logarithmic functions Graphing Logarithmic Functions
Graphing logarithmic functions
- We are asked to graph y is equal to log base 5 of x
- and just to remind us what this is saying,
- this is saying that y is equal to the power that I have to raise 5 to, to get to x.
- Or if i would write this logarithmic equation as an exponential equation
- 5 is my base
- y is the exponent that i have to raise my base to,
- and then x is what i get when I raise 5 to the yth power.
- So anther way of writing this equation would be
- 5 to the yth power is going to be equal to x.
- These are the same thing.
- Here we have y as a function of x, here we have x as a function of y.
- But they're really saying the exact same thing:
- "Raise 5 to the yth power to get x".
- But when you put this as a logarithm, you are saying:
- "To what power do I have to raise 5 to, to get x?
- Well, I have to raise it to y."
- Here, what do I get when I raise 5 to the yth power? I get x.
- Now that that is out of the way,
- let's make ourselves a little table
- that we can use to plot some points,
- and we can connect the dots to see what this curve looks like.
- So let me pick some Xs and some Ys.
- And in general we want to pick some numbers
- that give us some nice, round answers.
- Some nice, fairly simple numbers for us to deal with
- so we don't have to get a calculator.
- And so in general, you want to pick x values,
- you want to pick x values where the power that you have to raise 5 to to get that x value
- is a pretty straightforward power.
- Or another way to think about it is, you could just think about the different y values
- that you want to raise 5 to the power of
- and then you can get you x values.
- So we could actually think about this one to come up with our actual x values.
- But we want to be clear that when we express it like this,
- the independent variable is x and the dependent variable is y.
- We might just look at this one to pick some nice x's that give us nice, clean answers for y.
- So what happens, here I'm actually going to fill out the y first
- Just so that we get nice, clean x's.
- So let's say that we are going to raise 5 to the --I'm going to pick some new colours--
- to the negative 2 power, -- and let me do some other colours --
- negative 1, zero, 1, and I'll do one more, and then 2.
- So once again, this is a little nontraditional
- where I'm filling in the dependent variable first,
- but the way that we have written it over here, ......
- it's easy to to find out what the independent variable must be for this logarithmic function.
- So, what x gives me the y of negative 2?
- What does x have to be for y to be equal to -2?
- Well, 5 to the negative 2 power is going to be equal to x,
- so 5 to the negative 2, is 1 over 25, so we get 1/25.
- So another way, if we go back to this earlier one,
- if we say log, base 5, of 1/25.
- What power do I have to raise 5 to to get 1/25?
- Well, I have to raise it to the negative 2 power.
- Or you could say 5 to the negative 2 is equal to 1/25.
- These are saying the exact same thing.
- Now, let's do another one.
- What happens when I raise 5 to the negative 1 power?
- Well, I get one fifth. For this original one over there,
- we are just saying that log base 5 of 1/5, you want to be careful
- this is saying: "what power do I have to raise 5 to, in order to get 1/5?"
- Well, I have to raise it to the negative 1 power.
- Here, what happens when I take 5 tot the 0 power? I get 1.
- And so this relationship is saying the same thing as log, base 5, of 1,
- what power do I have to raise 5 to to get 1?
- Wel, I've just got to raise it to the 0 power.
- Let's... what happens when I raise 5 tot the first power?
- Well, I get 5.
- So if you go over here, that is just saying, what power do I have to raise 5 to to get 5?
- Well, I have to just raise it to the 1st power.
- And then finally, if I take 5 squared, I get 25.
- So if you look at it from the logarithm point of view, you say
- what power do I have to raise 5 to to get to 25?
- Well, I have to raise it to the second power.
- So, I kind of took the inverse of the logarithmic function. I wrote it as an exponential function.
- I switched the dependent and independent variables.
- So I could pick, or derive, nice clean x's that would give me nice, clean y's.
- Now, with that out of the way, but I do want you to remind,
- I could have just picked random numbers over here,
- but then I probably would have gotten less clean numbers over here, so I would have had to use a calculator.
- The only reason why I did it this way, is so I get nice clean results that I can plot by hand
- So let's actually graph it.
- So the y's go between -2 and 2,
- the x's go from 1/25 all the way to 25
- So let's graph it.
- So that is my y-axis, and this is my x-axis.
- So I'll drw it like that, that is my x-axis and then the y's
- start at zero and then you get to positive 1, positive 2,
- and then you have -1, -2 and then on the x-axis it is all positive
- and I'll let you think about whether the domain here is, well we can think about it,
- Is a logarithmic function defined for an x that is not positive?
- Is there any power that I could raise 5 to so that I would get 0?
- No. You could raise 5 to an infinitely negative power to get a very very small number
- that approaches 0, but you can never get-
- there is no power that you can raise 5 to to get 0.
- So x cannot be 0. There is no power that you can raise 5 to
- to get a negative number. So x can also not be a negative number.
- So the domain of the function right over here-
- and this is relevant becausewe want to think about what we are graphing -
- The domain here is x has to be greater than 0.
- Let me write that down.
- The domain here is that x has to be greater than 0.
- So we are only going to be able to graph this function in the positive x-axis.
- So with that out of the way, x gets as large as 25,
- so let me put those points here, so that's 5, 10, 15, 20
- and 25.
- And then let's plot these.
- ... and is blue and x is 1.25 and y is -2.
- When x is 1/25, is it going to be really close to there, then y is negative 2.
- So that's going to be right over there.
- Not quite at the y-axis, 1/25 .. of the y-axis, but pretty close.
- So that right over there is 1/25 comma -2 right over there.
- Then when x is 1/5, which is slihtly further to the right,
- 1/5 with y = -2. So right over there.
- This is 1/5, -1. And then when x is 1, y is 0.
- So 1 might be there, so this is the point (1,0).
- And then when x is 5, y is 1.
- I;ve covered it over here, y is 1.
- So that's the point (5,1).
- And then finally when xis 25, y is 2.
- So this is (25,2). And then I can graph the function.
- And I'll do it in the colour pink.
- So as x get's super super super small, y goes to negative infinity.
- So what power hve you have to rais e5 to to get point .0001
- That has to be a very negative power.
- So we get very negative as we approach 0,
- and then it kind of moves up like that.
- And then starts to curve to the right like that.
- And this thing right over here is going to keep going down at an ever steeper rate
- and it's never going to quite touch the y-axis.
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