The relationship between

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the relationship between

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If we're assuming that b is not 0, if we're assuming that b is not 0, so we're going to assume that, and we can assume, and I think that's a safe assumption because where we're raising b to all of these other powers, we're getting a non-0 value.

Since we know that b is not 0, anything with a 0 power is going to be 1. This tells us that a is equal to 1. We got one figured out. Now let's look at this next piece of information right over here. What does that tell us? That tells us that log base b of 2 is equal to 1.

the relationship between

This is equivalent to saying the power that I needed to raise b to get to to 2 is 1. Or if I want to write in exponential form, I could write this as saying that b to the first power is equal to 2.

I'm raising something to the first power and I'm getting 2? What is this thing? That means that b must be 2.

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So b is equal to 2. You could say b to the first is equal to 2 to the first. That's also equal to 2. So b must be equal to 2. We've been able to figure that out. This is a 2 right over here. It actually makes sense.

the relationship between

Now let's see what else we can do. Let's see if we can figure out c. Let's look at this column. Let's see what this column is telling us. That column we could read as log base b. Now our y is 2c. Log base b of 2c is equal to 1.

Or we could read this as b, if we write in exponential form, b to the 1. Now what's b to the 1.

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They tell us right over here that b to the 1. We get 2c is equal to 3, or divide both sides by 2, we would get c is equal to 1. This is working out pretty well. Now we have this last column, which I will circle in purple, and we can write this as log base b of 10d is equal to 2.

This is saying the power I need to raise b to to get to 10d is 2. Now what is b to the 2. 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.

the relationship between

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 The power I need to raise b to, to get to 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.

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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.