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Relationship between exponentials & logarithms: graphs

CCSS.Math:

Video transcript

Voiceover: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. I've actually copy and pasted this problem on my little scratch pad so I can mark it up a little bit. What is this first function? This first function is telling us so x, and this is y is equal to b to the x power. When x is equal to zero, y is equal to one. That's this point right over here. When x is equal to one, b to the one power or b to the first power is equal to four. y is equal to four. Another way of thinking of this y or four is equal to b to the first power and actually we can do [said] must be four. That's this point right over there. This point is telling us that b to the second power is equal to 16. When x is equal to two, b to the second power, y is equal to 16. Now we want to plot the three corresponding points on this function. Let me draw another table here. 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. What are the possibilities here? What I want to do is think ... Let's take these values because these are essentially inverse functions log is the inverse of exponents. If we take the points one, four, and 16. What is y going to be here? y is going to be log base b of one. This is saying what power I need to raise b to to get to one. If we assume that b is non zero and that's a reasonable assumption because b to different powers are non zero, this is going to be zero for any non zero b. This is going to be zero right there, over here. We have the point one comma zero, so it's that point over there. Notice this point corresponds to this point, we have essentially swapped the x's and y's. In general when you're taking an inverse you're going to reflect over the line, y is equal to x and this is clearly reflection over that line. Now let's look over here, when x is equal to four what is log base b of four. What is the power I need to raise b to to get to four. We see right over here, b to the first power is equal to four. We already figured that out, when I take b to the first power is equal to four. This right over here is going to be equal to one. When x is equal to four, y is equal to one. Notice once again, it is a reflection over the line y is equal to x. When x is equal to 16 then y is equal to log base b of 16. The power I need to raise b to, to get to 16. Well we already know, if we take b squared, we get to 16, so this is equal to two. When x is equal to 16, y is equal to two. Notice we essentially just swapped the x and y values for each of these points. This is y, this is a reflection over the line y is equal to x. Now, let's actually do that on the actual interface. The whole reason is to give you this appreciation that these are inverse functions of each other. Let's plot the points. That point corresponded to that point, so x zero, y one corresponds to x one, y zero. Here x is one, y is four that corresponds to x four, y one. Here x is two, y is 16 that corresponds to x 16, y is two. We got it right.