Now some facts about or the essential characteristic about the exponential

function is that the rate of change of y is proportional to y itself.

And what that tells you is that there's an interpretation in the background here

of m for small values, again these are approximations for these interpretations.

So let's say m is a small number, for example, between -0.2 and 0.2.

Then what's going to come out of the exponential function is the idea that for

every one-unit change in x,

there's going to be an approximate 100m% proportionate change in y.

So what you're seeing in the exponential function, and

it's differing from the power function,

is now we're talking about absolute change in x being associated with percent, or

proportionate change in y, and we're claiming that that is a constant.

You go back to the power function, we were looking at percent change in x,

relating to percent change in y through the constant m.

And if we go back to the linear function, we were seeing absolute change

in x being related to absolute change in y through the constant m.

So.these different functions that we're looking at are capturing how

we're thinking about x and y changing.

Are we thinking about them changing in an absolute sense, or

are we thinking about them changing in a relative sense.

So just going back to this interpretation here of the constant m in

the exponential function, we can say for example if m = to 0.05, then

a one-unit increase in x is associated with an approximate 5% increase in y.

And that 5% is constant, is doesn't matter or the value of x.

So every time x goes up by one unit, y increases approximately

by another 5%, a relative or proportionate change.

So once again the exponential function lets us understand

how absolute changes in x are related to relative changes in y.

One more to go and that's the log function.

This is the log transformation.

It's probably the most commonly used transformation in quantitative modeling.

We're not looking at the raw data then often times we're looking at

the log transform of the data and this is what a log curve looks like.

It's an increasing function but

the feature is that it's increasing at a decreasing rate.

So the log function is extremely useful

when it comes to modeling processes that exhibit diminishing returns to scale.

So diminishing returns to scale, says we're putting more into the process.

But each time we put an extra thing into the process, yeah,

we get more out but not as much as we used to.

And so you might think of diminishing returns to scale as you've

cooked a big meal at Thanksgiving and it needs to be cleaned up.

Now if you're doing the clean up by yourself, that takes quite a while.

If you have one person help you, it's probably going to be a bit faster,

and maybe you had two people help you, it's going to be even faster.

But if you go up to ten people in the kitchen all trying to help you clear up

that meal, at some point people start getting in the way of one another.

And the benefits of those incremental people coming in to help you clear up,

really fall away quite quickly.

And so that's an idea of diminishing returns to scale.

From a mathematical process point of view,

we think about the log function as increasing but at a decreasing rate.

Now, as I said, all of these functions that I'm introducing have essential

characteristics.

And the essential characteristic of the log function is that a constant

proportionate change in x is associated with the same absolute change in y.

So notice how that's the flip side of the exponential function?

The exponential function had absolute changes in x being

related to relative changes in y.

The log function is doing it the other way around.

We're talking about Proportionate changes in x being associated with the same

absolutely changing y.

Again, when you get to the stage of doing modeling and

you're thinking about the business process, you need to be thinking about

these ideas as you choose your model functional representation of the process.

How do you think things are changing?

Do you think it's absolute change in x being related to absolute change in y

as a constant?

Or do you think it's relative change in x to relative change in y.

You think it's relative change in x to absolute change in y or

absolute change in x to relative change in y?

And here in the log function, again the essential characteristic that constant

proportionate changes in x are associated with the same absolute change in y.

If you think your business process looks like that then the log function

is a good candidate for a model.