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Find an exponential function determined by the data in Table 7

$$23.5294(0.85)^{x / 8} \approx 23.5294(0.97989)^{x}$$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 2

Exponential Functions

Campbell University

Oregon State University

Harvey Mudd College

Lectures

02:51

Find an exponential functi…

02:45

03:25

01:35

Find the exponential funct…

01:05

Determine the exponential …

01:29

01:06

So if we want to find an exponential equation that represents this data here, we can first start by just looking at why is a b to the X plug two of these points in, uh and then go from there. So depending on which two points you pick, you might get something slightly different from what I'm going to get. Um, but as long as the numbers are kind of close to each other that I would say it's probably about the same. But then again, it may just give drastically different answers as well. But if it does, that's still okay, because we're just using different points. Um, so I'm just gonna use the first two points. So what we're gonna do is just plug these in, so I'm first gonna plug in eight, and it should have an output of 20. So B 20 is equal to a B, um, to the eight, and then plug in 16, so that should be equal to 17 will be 17 a times B to the 16th. Now, you could go ahead and solve for a set them equal to each other. But what I like to do is divide these equations because then a just goes away and I really don't have to think that hard. So on the left side will have 17/20 is equal to, and then B to the 16 over B to the eight is just going to be be to the eight. And now we can take the eighth power on each side. And that just gives B is equal to 17/20 raised to the 1/8. Uh huh. Now, in order for us to solve for pay, we can just take this and plug it into any of the equations. I'm just gonna plug it into the second one, so that would give us 20 is equal to a and then we have 17/20 raised to the eighth, and then we have the eight power. So those canceled out, and then we would just need to multiply by 20 and divided by 17, so that would be 400 over 17 is equal pay. So now we can just plug that into our equation we started with would be why is equal to 400 over 17 and then 17/20 raise to the 1/8 power and then raised to the X. Um so you could actually distribute that X if you want it, and then just have it 400 over 17 time, 17/20 raised to the X over eight like that so you can write it either way. But obviously the like this since this way, it looks a little bit prettier, I would say. And again, Um, the other points may give you a slightly different answer. Like I said, I haven't checked those kind of confirm it. But since they just said give one possible answer, I figured I would just go ahead and use the first two.

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