Graphical relationship between 2ˣ and log₂(x)

what I want to do in this video is graph up a classic exponential function and then graph a related logarithmic function and see how the two are related visually and the two things I'm going to graph are that Y is equal to 2 to the X power and Y is equal to the log base 2 of X and I encourage you to pause the video make a table for each of them and try to graph them on the same graph paper and see how they are related and if you see how they're related think about why they are related that way so let's first start with y equals 2 to the X power so I'll make a little table here different X values and the corresponding Y values so x and y we could start with negative 2 negative 1 0 1 2 3 so in each case Y is going to be 2 raised to this power so 2 to the negative 2 power is going to be 1/4 2 to the negative 1 power is 1/2 2 to the 0 power is 1 to the first power is 2 2 to the second power is 4 2 to the 3rd power is 8 so let's graph that so 2 to the third power is 8 2 to the fourth 2 to the 2 to the 4 so I 2 to this 2 to the third power is 8 2 to the second power is 4 2 to the first power is 2 2 to the zeroth power is 1 2 to the negative 1 power is 1/2 2 to the negative 2 power is 1/4 we'd even a 2 to the negative third power is going to be 1/8 so it's going to look something like this so the graph is going to look it's going to look something it's going to look something like this right over here and it's kind of your classic sometimes this will be called your exponential hockey stick because it kind of looks like a hockey stick or just it just kind of starts kind of slow and just BAM shoots straight up and notice as we go to the left as X becomes more and more and more negative our value approaches 0 but never quite gets there if we have 2 to the negative 1 millionth power it's going to be a very very small number very very close to zero but it's not going to be quite zero so it's going to we're going to have a horizontal asymptote at X or at Y is equal to zero or the x-axis is a horizontal asymptote fair enough now let's graph now let's graph y is equal to log base two of X and before I graph that let's just think about another way of representing it this literally says for any X what power what exponent Y would if I raise to two that would give me X so this is an equivalent statement as saying two to the Y power is equal to X and if you notice what we've done here between these two things you're essentially just switching the X's and Y's here it's two to the X power is equal to Y here's two to the Y power is equal to X so really this and this you've swapped the X's and the Y's so you've swapped the XS and the Y's and what we will see is that we can essentially swap these two columns so x and y so let me just do 1/4 1/2 1 1 2 4 and 8 2 4 and 8 so here now we're saying if X is 1/4 if X is 1/4 what power do we have to raise 2 to to get to 1/4 we have to raise it to the negative 2 power 2 to the negative 1 power is equal to 1/2 2 to the 0 power is equal to 1 2 to the first power is equal to 2 2 to the second power is equal to 4 2 to the third power is equal to 8 notice all we did is we essentially swapped these two columns and so let's graph this when X is equal to 1/4 y is equal to negative 2 when X is 1/2 Y is equal to negative 1 y is equal to negative 1 when X is 1 Y is 0 when X is 2 y is 1 when X is 4 y is 2 when X is 8 y is three so that's going to look like this so notice I think you might already be seeing a pattern right over here these two graphs are essentially the reflections of each other and what would you have to reflect about to get these two well you'd have to reflect about Y is equal to X so if you swap the XS and the Y's so so another way to think about if you swap the axis you would get the other graph - essentially what we're - essentially what we're doing notice it's symmetric about that line and that's because these are essentially the inverse functions of each other one way to think about it is we swapped we swapped the X's and Y's just as this as as X becomes more and more and more and more negative you see Y approaching 0 here you see as Y is becoming more and more negative as Y is approaching is getting more and more negative X's approaching 0 or you could say as X approaches 0 Y becomes more and more and more negative so the whole point of this is just to give you an appreciation for the relationship between an exponential function and a logarithmic function they're essentially inverses of each other and you see that in their graphs they're reflections of each other about the line y is equal to X