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 positive 0.2.

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

every 1 unit change in x,

there's going to be an approximate 100 times m% 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 = 0.05, then a one-unit increase in x is associated

with an approximate 5% increase in y and that 5% is cosmic, it doesn't matter.

Or the value of x.

So every time x goes up by 1 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, you 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 cleanup by yourself, it takes quite a while.

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

Maybe if 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 dimensions returns from 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 an 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

absolute change in 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, a 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?

Do 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 proportion that changes in x are associated with the same

absolute changes in y.

If you think your business process looks like that,

then the log function is a good candidate for a model.