[music]. Welcome.

In this unit, you'll be learning about exponential and logarithmic functions.

You've already spent a great deal of time learning about functions.

But these two functions are so special that they're worth their own unit.

These functions get used a lot in applications.

So let's start with an application to see why we need an exponential function and a

logarithmic functions, and why the ones you've learned so far won't suffice.

Let's begin by looking at an example to understand why we need the exponential and

logarithm functions. Suppose we have a bunny.

And if this bunny has a bunny every month, well after one month, we'd have 1 bunny.

After 2 months, he would have a bunny, and we would have 2 bunnies.

After 3 months, each of those 2 bunnies would have a bunny.

And we'd have 4 bunnies. If each of those 4 bunnies had a bunny, we

would have 8 bunnies, and so on. And eventually, we'd have a ton of

bunnies. So suppose we track the bunny's population

over time. What if we wanted to know how many bunnies

there were after a month, after two months, after five months, after a year,

after five years, after a century? To do these sorts of calculations, we

would need to have some sort of function which capture the doubling behavior of the

bunny population. So the function we're going to look at is

something called the exponential function. The, we see we have a graph and in this

graph, we see The population of the bunnies for several iterations.

We started with 1 bunny, then we had 2, then we had 4, and so on.

What kind of function matches this behavior?

Well if you look back at the, think about the catalog of functions you already know.

You know linear functions, polynomial functions, rational functions, and you try

to think about this curve over here. Well what does that look like?

This curve doesn't look like any of the curves you're familiar with.

You might guess a parabola, because it seems to be curving up.

However, that won't work for this population.

If we want to match this with a curve, it would have to look something like this.

And that curve is getting steeper and steeper, because this population of

bunnies is getting large, very, very quickly.

And that brings us to the need for the exponential function.

In John Napier's book, a wonderful description of logarithms.

He described how the logarithm was such a useful tool to help do calculations for

celestial mechanics. This book was the first one of its time to

describe how a logarithm could be used to turn hard multiplications into simple

additions. He also gave a table of logarithmic

values, which was used for hundreds of years afterwards.

Another mathematician Leonard Euler played a large role in the development of the

exponential and logarithm functions. Leonard Euler was actually the first guy

to use the notation for a function f of x. So whenever you see the f parentheses x

that was due to Euler. Euler was a very prolific mathematician.

He wrote over 800 published papers in his life time and his works would complete 90

volumes. In one of Euler's works he looked at a

constant that came up from numerous calculations, and it's since been known as

Euler's constant, or Euler's number. Let's look a little more closely now at

Euler's constant. Euler's constant, denoted by an e, is

equal to 2.712818 etcetera. Eulers constant is an infinite

nonrepeating decimal number, which we call a transcendental number in mathematics.

You're already familiar with the transcendental number pi, 3.1415, et

cetera, that's a number you've probably seen all ready seen in your studies.

Euler's constant can be thought of as coming from a wide variety of areas,

including things like geometry, the interest when you're talk about interest

on a loan or interest you earn in a bank account and other situations such as

continued fractions. Euler's constant can be derived from a

number of these situations, and you'll encounter these more later in your

mathematical studies. Let's move on and talk about the function

that results from Euler's constant, or the exponential function.

Using that notation that Euler developed for functions, f of x is equals e to the x

is called the exponential function. This means we take that Euler constant

number 2.71 et cetera and raise it to any number x as the power.

For example f of 1 would just be e raised to the 1 power which is just e itself or

that 2.71 et cetera. The related function, the, the exponential

function is a logarithm function. Here you see our logarithm function g of x

equals log base e of x. We also denote this as ln.

The ln stands for natural logarithm, or it is special logarithm function where the

base is e. The base is that little number written as

a subscript of the logarithm. We read this as log base e of x.

These two functions are very closely related to each other, namely, they're

inverse functions, for each other. Inverse functions have the property that,

when we compose them together, they undo each other, and we're just left with x.

For example, a composition, f of g of x, or e to the ln of x, just gives me x.

Same thing if I compose them in the opposite order, g of f of x equals ln of e

to the x power and those exponentials and logs again undo each other and I'm just

left with x. That's the inverse property of functions.

You may have just seen, we did this with the base of e.

E isn't the only base you can use when dealing the exponential and logarithm

functions. Let's look now at the general exponential

and logs. The general exponential function would

just be f of x equal to a to the x. A here is taking the place of some

constant Instead a could be any positive number not equal to 1, so is got to be

greater than 0 but we wont let it be equal to 1 the reason is if a was 1, 1 to any

power is just 1 and we get a constant function which really doesn't satisfy the

properties of the exponential functions that we will be talking about shortly,

another function we can look at g of x equals log base a of x Again, that

subscript on the log stands for the base. We can convert back and forth between the

logarithm and the exponential function according to if y equals log base a of x.

This really just means that a to the y power equals x.

X. Once again, these two functions have the

inverse property. If I compose them together, f of g of x or

g of f of x, the exponential and logarithm undo each other and I'm just left with the

x. Let's do a quick example so you can see

how the log works because with different bases, sometimes this can be new to

students. For example, what if I want to compute the

log base 2 of 16? First I'm going to notice to myself that 2

to the 4th power, or 2 times 2 times 2 times 2, is just equal to 16.

So 2 raised to the 4th power is 16. In other words log base 2 of 16 is asking

the question, what power do I need to raise 2 to, to get 16.

And the answer is 4 because 2 raised to the 4th power gives me 16.

So you can see logarithm is kind of asking the inverse question of, what power do I

need to raise the base to, to get the quantity that I'm taking the logarithm of?

Let's review some of the exponential properties that you'll see later in this

course. You see a lot of properties listed here.

You'll be talking about these in much more detail later on.

But for right now, what you'll want to be noticing is that there's a lot of

properties of exponential functions, and utilizing these will allow you to solve

equations involving exponents, specifically exponential equations with e,

or any other base that we're talking about.

A and b here are standing for bases. Remember those numbers have to be

positive, and cannot equal 1. The logarithm also has a lot of

properties. Don't worry about memorizing these now.

But you will be utilizing them later to solve logarithm equations.

Some of these properties come up again and again in calculus.

So I highly recommend you pay attention now.

Learn this well because it'll help you a lot later in your calculus studies.

We can also consider the properties of the graphs of exponentials and logarithms.

Here I've shown you an example of an exponential graph.

This is if the value of a or the base of the exponential is greater than 1.

If the value were between zero and one, the function would just be Going down to

the right, and increasing on the left. That would be a negative, a base that's

smaller. And it would be negative growth or

decreasing function. Notice here, I've listed some properties

of the exponential function graph. 1 of the key properties is all exponential

functions go through this special point 0, 1.

This is because anything raised to the 0th power just gives you a 1, and that's why

that special point is on all of our exponential functions.

Another property you'll notice from the graph is this function is continuous.

There's no gaps, breaks, or jumps in our curve.

This also has a horizontal asymptote at the x axis, or y equals 0.

Notice, as I go to smaller values of x, the function gets closer and closer to the

x axis, but it actually never reaches it and that's why it's an asymptote.

We also have the properties that this function increases as I go to the right,

and decreases as I go to the left. Finally notice this function is 1 to 1.

Remember we checked for functions being 1 to 1 by looking at the horizontal line

test. Any horizontal line across my curve will

intersect the graph at exactly one point and not more than one point.

Alright let's now look at the exponential functions inverse namely the logarithm

function for contrast. The logarithm function here is given by

the graph like this. A logarithm function graph is also

continuous because notice there's no breaks gaps or jumps.

It goes through the special point 1, 0. It also has a vertical asymptote at x

equals 0, or the y axis. Notice that this function also increases

as we go to the right. And as I head towards x equals 0, this

function goes to negative infinity. This function is also 1 to 1.

Because, notice, any horizontal line through this function.

Will intersect the graph, at most, 1 time. How are these two functions related to

each other? Well, if you start to look at their

graphs, you can probably see that these 2 functions are inverses, which we've been

talking about a lot. And if we look at the graphs, you can see

that more easily. So here, I've graphed a sample of an

exponential function with base e. And a logarithm function, or natural

logarithm, ln of x. The red line there is the line y equals x.

Notice if I folded the screen over the red line, the 2 curves would line up over each

other. That's because they're inverse functions

so they're basically a reflection of across the line y equals x.

Well, let's see what are we going to learn about exponentials and logarithms in this

course. In this unit, you will learn to, first of

all, evaluate exponential and logarithm expressions.

You'll also learn to convert back and forth between the exponential and

logarithm form of an equation. You'll learn to graph those exponential

and logarithm functions. Particularly when you have

transformations. You'll learn to solve exponential and

logarithm equations. That's going to be one of the most

important skills that you'll need to take with you to your calculus course.

Finally, we'll talk about using exponentials and logarithms to solve

application problems. Speaking of applications, what are the

applications of exponentials and logarithms?

There's quite a few applications including, population dynamics is probably

the most standard one you'll hear about. If we look at the population for example,

the bunnies we looked at earlier growing with time you'll often see exponential

growth. Radioactive decay is another popular

application of exponentials and logarithms.

You may have heard of carbon dating. Carbon dating is where they figure out the

age of old artifacts using the decay of carbon over time And that's looking at an

example of decay, in this case it's not radioactive.

But we are looking at the decay of a substance.

Also, if you're ever investing money in the bank, you're often getting compound

interest. Compound interest is an example of an

application of exponential functions. Newton's law of cooling is the law that

tells you, if you put a cake on the counter, how quickly that cake will reach

room temperature or if you put hot soda in the fridge, how quickly will that soda

cool. That's Newton's law of cooling and that is

another application of exponential and logarithms.

Finally, the Richter scale for earthquakes in California, you may be very familiar

with this. The Richter scale for earthquakes is

actually a logarithm function. And those numbers that tell you how severe

an earthquake is, is actually based on a logarithmic scale.

Sound intensity is another thing that is measured in logarithms.

The sound intensity decibel levels, is a logarithmic funciton.

Function. And finally the learning curve, which is

the rate at which you acquire knowledge can be modeled closely by exponential and

logarithm functions. So if you're looking to figure out how

much information you retain as a function of time at which you are studying, that

actually will be modeled by exponentials and logarithms.

Well, I hope you enjoyed learning about expoential and logarithm functions.

Thank you, and I'll see you next time. [music]