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# Graphs of logarithmic functions

CCSS.Math:

## Video transcript

so we have a graph right over here and we have four potential function definitions for that graph and so what you might want to do is pause the video right now and think about which of these function definitions are actually being depicted in this graph right over here so I'm assuming you've given a go at it now let's work through it together before we even address these we all see that they all have a log base two in the in the function definition let's just remind ourselves what y equals log base 2 of X even looks like and we could think about well what happens if we were to add one or subtract one from it or if we were to shift it a little bit so let's just think about some interesting values here let's think about some interesting values let's think about what happens so I'll have x and y let's think about when X is equal to two I picked two because that's if X is equal to two years saying well log base two of to what power do we have to raise 2 to to get to 2 we'll have to raise it to the first power what about when X is equal to actually let me try let me do do several let's do when X is equal to 8 log base 2 of 8 is 3 I raise 2 to the third power I get 8 let's do 4 log base 2 of 4 is 2 2 to the second power is equal to 4 let's do 2 where we started off with well that log base 2 of 2 is going to be 1 2 to the first power is equal to 2 now let's think about when X is equal to 1 what I have to raise 2 to to get 2 1 well I raise it to the 0th power 2 to the 0th power is equal to 1 now let's think about how we would get to how we would get to 1/2 what do I have to raise 2 to to get to 1/2 well 2 to the negative 1 power is going to be equal to 1/2 and I can keep going what about 1/4 well that's 2 to the negative 2 power is 1/4 I could go to 1/8 2 to the negative 2 to the negative 3 power is equal to 1 now let's just graph some of these points when X is equal to 8 y is equal to 3 when X is equal to 4 y is equal to 2 when X is equal to 2 y is equal to 1 when X is equal to 1 Y is equal to 0 I think you see the general shape already forming when X is 1/2 Y is negative 1 when X is 1/4 Y is negative 2 thick you see where this is going when X is 1/8 Y is negative 3 and so you have a graph that looks something like this looks something like this I'm just connecting the dots and this is the behavior this is the behavior that we that we would expect as X becomes really really really large you think about what power do I have to raise 2 to to to to get that X well it's going to increase but it's going to increase in an ever decreasing rate and then we see that as X approaches 0 from the right as to get closer and closer to 0 you have to raise to to more and more and more negative values so the log as we approach is 0 get becomes very very very very very negative and we can never quite get to x equals 0 if you put an actual 0 right over here what power would you have to raise 2 to to get to 0 well you can't you can get close to 0 by raising 2 to a very very very very negative a very very negative value and this thing right over here it's not going to be even defined for negative well for any for non positive X's so that's why we have it not defined for anything less than or equal to 0 this domain right over here is only for positive X's so that's log base 2 of X how is this thing right over here how does this thing right over here look different well the obvious thing that jumps out at me is that well it's flipped over the x-axis so that's a pretty good sign that this is going to we're going to have a negative log base two of X so let's actually now graph let's let's graph that what's Y what's y is equal to negative log base two of X look like well each of these points we're just going to flip it over we're just going to flip it over the x-axis so we're going to go there we're going to go there we're going to go there and then this this is going to go through that point right over here and then and then let's see instead of it 1/2 it's going to be like this so y equals negative log base 2 of X is going to look something like this maybe see if I can draw it neatly so it's going to look it's going to look something something like this something something like this so we're getting close to the blue graph so that's y equals negative log base two of X so what's the difference between the green graph and the blue graph well if you look carefully you see you see that the blue graph is essentially the green graph shifted to the left by two shifted to the left we've shifted it to the left by 2 at every one of these points at every one of these points we have shifted to the left by 2 so how do you shift to the left by 2 well you would replace the X with an X plus 2 so you would replace the X with an X plus 2 and one way to think about it is well in the original negative log base 2 of X when we see this asymptote at x equals 0 now we're going to see the asymptote when this whole expression X plus 2 is equal to 0 X plus 2 is equal to 0 when X is equal to negative 2 when X is equal to negative 2 you see the asymptote right over there so I encourage you could try values out if you like you can essentially take all of these values you could take all of these values and subtract 2 from them and then when you add two you're going to get back to these values so if you shift to the left by 2 that's like replacing the X with an X plus 2 so this is what is being graphed and once again try out the values if if you don't believe me