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Current time:0:00Total duration:4:10

Relationship between exponentials & logarithms: graphs

CCSS.Math:

Video transcript

the three points plotted below are on the graph of y is equal to B to the X power based only on these three points plot the three corresponding points that must be on the graph of y is equal to log base B of X by clicking on the graph so I've actually copied and pasted this problem on my little scratch pad so I can mark it up a little bit so what is this this first function this first function is telling us so X and this is y is equal to B to the X power so when X is equal to zero Y is equal to one when X is equal to zero Y is equal to one that's this point right over here when X is equal to one when X is equal to 1 B to the 1 power or B to the first power is equal to 4 y is equal to 4 so another way of thinking this Y or 4 is equal to B to the first power and actually we can deduce then that B must be 4 so that's this point right over there and then this point is telling us that B to the second power is equal to 16 so when X is equal to 2 B to the second power Y is equal to 16 now we want to plot the three corresponding points on this function so let me let me draw another table here so now it's essentially the inverse function where this is going to be X and we want to calculate Y is equal to log base B of X and so what are the possibilities here so what I want to do is think let's take these values because these are the ones these are essentially inverse functions log is the inverse of exponents so if we take the point 1 4 1 4 and 16 1 4 and 16 1 4 and 16 so what is Y going to be here Y is going to be log base B of 1 so this is saying what power I need to raise B to to get to 1 well if we assume that B is nonzero and that's a reasonable assumption because be two different powers are nonzero this is going to be 0 for any nonzero be so this is going to be zero right they over here so we have the point 1 comma 0 so it's that point over there and notice this point corresponds to at this point we've essentially swapped the X's and Y's and in general when you're taking an inverse you're going to reflect over over the line y is equal to X and this is clearly a reflection over that line now let's look over here when X is equal to 4 what is log base B of 4 what is the power I need to raise B to to get to 4 well we see right over here B to the first power is equal to 4 we already figured that out when I take B to the 4 B to the first power is equal to 4 so this right over here is going to be equal to 1 so when X is equal to 4 y is equal to 1 and notice once again it is a reflection over the line y is equal to X and so where when X is equal to 16 when X is equal to 16 then Y is equal log base B of 16 the power I need to raise B to to get to 16 well we already know if we take if we take B squared we get to 16 so this is equal to 2 so when X is equal to 16 y is equal to 2 notice we've essentially just swapped the X and y values for each of these points which is why this is a reflection over the line y is equal to X now let's actually do that on the actual interface and the whole reason is is to give you this appreciation that these are inverse functions of each other so let's plot let's let's plot the points so that point corresponding to that point so X 0 y 1 corresponds to x 1 y 0 here X is 1 Y is 4 that corresponds to X 4 y 1 here X is 2 y is 16 that corresponds to X 16 y is y is 2 and we got it right