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Shape of a logarithmic parent graph

Sal graphs y=log₅(x). Created by Sal Khan and Monterey Institute for Technology and Education.

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Video transcript

We're 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 were to 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 another way of writing this equation would be 5 to the y'th 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 y'th power to get x. When you put it as a logarithm, you're saying, well, what power do I have to raise 5 to to get x? We'll have to raise it to y. Here, what do I get when I raise five to the y power? I get x. That out of the way, let's make ourselves a little table that we can use to plot some points, and then we can connect the dots to see what this curve looks like. So let me pick some x's and some y's. And we, in general, want to pick some numbers that give us some nice round answers, some nice fairly simple numbers for us to deal with, so that we don't have to get the calculator. And so in general, 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, you could just think about the different y values that you want to raise 5 to the power of, and then you could get your 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 even or nice x's that give us nice clean answers for y. So here, I'm actually going to fill in the y first, just so we get nice clean x's. So let's say we're going to raise five to the-- let's say we're going to raise it-- I'm going to pick some new colors-- negative 2, negative 2 power-- and let me do some other colors-- negative 1, 0, 1. 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've written it over here, it's actually given the dependent variable, it's easy to figure out what the independent variable needs to be for this logarithmic function. So, what x gives me a y of negative 2? What x gives me-- what does x have to be for y to be equal to negative 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 over 25. If we go back to this earlier one, if we say log base 5 of 1 over 25, what power do I have to raise 5 to to get 1 over 25? We'll have to raise it to the negative 2 power. Or you could say 5 to the negative 2 is equal to 1 over 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? I get one fifth. So if we go to this original one over there, we're just saying that log base 5 of one fifth. Want to be careful. This is saying, what power do I have to raise 5 to in order to get one fifth. We'll have to raise it to the negative 1 power. What happens when I take 5 to the 0'th power? I get one. And so this relationship-- This is the same thing as saying log base 5 of 1. What power do I have to raise 5 to to get 1? I just have to raise it to the 0th power. Let's do the next two. What happens when I raise 5 to the first power? Well, I get 5 So if you go look over here, that's just saying, log, what power do I have to raise 5 to to get 5? We'll have to just raise it to the first power. And then finally, if I take 5 squared, I get 25. So when you look at it from the logarithmic point of view, you say, well, what power do I have to raise 5 to to get to 25? We'll have to raise it to the second power. So I took the inverse of the logarithmic function. I wrote it as an exponential function. I switched the dependent and independent variables, so I can derive nice clean x's that will give me nice clean y's. Now with that out of the way, but I do want to remind you, I could have just picked random numbers over here, but then I would have probably gotten less clean numbers over here. 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. Let's actually graph this thing over here. So the y's go between negative 2 and 2. The x's go from 1/25th all the way to 25. So let's graph it. So that is my y-axis, and this is my x-axis. Draw it like that. That is my x-axis. And then the y's start at 0. Then, you get to positive 1, positive 2. And then you have negative 1. And you have negative 2. And then on the x-axis, it's all positive. And I'll let you think about whether the domain here is-- well, when you think about it-- is a logarithmic function defined for an x that is not positive? So is there any power that I can raise five to that I can get 0? No. You could raise five to an infinitely negative power to get a very, very, very, very small number that approaches zero, but you can never get-- there's no power that you can raise 5 to to get 0. So x cannot be 0. And there's no power then you could raise 5 to get another negative number. So x can also not be a negative number. So the domain of this function right over here-- and this is relevant, because we want to think about what we're graphing-- the domain here is x has to be greater than zero. Let me write that down. The domain here is that x has to be greater than 0. So we're 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 graph-- we put those points here. So that is 5, 10, 15, 20, and 25. And then let's plot these. So the first one is in blue. When x is 1/25 and y is negative 2-- When x is 1/25 so 1 is there-- 1/25 is going to be really close to there-- Then y is negative 2. So it's going to be like right over there, not quite at the y-axis. We're at 1/25 to the right of the y-axis. But pretty close. So that's right over there. That is 1 over 25, comma negative 2 right over there. Then, when x is one fifth, which is slightly further to the right, one fifth y is negative 1. So right over there. So this is one fifth, negative 1. Then when x is 1, y is 0. So 1 might be right there. So this is the point 1,0. And then when x is 5, y is 1. When x is 5, I covered it over here, when this is five, y is 1. So that's the point 5,1. And then finally, when x is 25, y is 2. So this is 25,2. And then I can graph the function. And I'll do it-- let me do it in a color-- I'll use this pink. So as x gets super, super, super, super small, y goes to negative infinity. It gets really small-- to get x's or as x becomes-- if you say what power do you have to raise 5 to to get 0.0001? It has to be very, very, very negative power. So y is going to be very negative as we approach 0. And then it kind of moves up like that. And then starts to kind of curve to the right like that. And this thing right over here, is going to keep going down at a steeper and steeper rate. And it's never going to quite touch. the y-axis. It's going to get closer and closer to the y-axis. But it's never going to be quite touch it.