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[music]. Welcome.

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

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

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

Â These functions get used a lot in applications.

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

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

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

Â logarithm functions. Suppose we have a bunny.

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

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

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

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

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

Â bunnies. So suppose we track the bunny's population

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

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

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

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

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

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

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

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

Â What kind of function matches this behavior?

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

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

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

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

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

Â However, that won't work for this population.

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

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

Â bunnies is getting large, very, very quickly.

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

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

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

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

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

Â additions. He also gave a table of logarithmic

Â values, which was used for hundreds of years afterwards.

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

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

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

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

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

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

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

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

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

Â equal to 2.712818 etcetera. Eulers constant is an infinite

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â functions. Let's look now at the general exponential

Â and logs. The general exponential function would

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

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

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

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

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

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

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

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

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

Â X. Once again, these two functions have the

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

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

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

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

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

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

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

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

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

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

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

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

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

Â course. You see a lot of properties listed here.

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

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

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

Â equations involving exponents, specifically exponential equations with e,

Â or any other base that we're talking about.

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

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

Â properties. Don't worry about memorizing these now.

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

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

Â So I highly recommend you pay attention now.

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

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

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

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

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

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

Â smaller. And it would be negative growth or

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

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

Â functions go through this special point 0, 1.

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

Â that special point is on all of our exponential functions.

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

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

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

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

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

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

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

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

Â test. Any horizontal line across my curve will

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

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

Â function for contrast. The logarithm function here is given by

Â the graph like this. A logarithm function graph is also

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

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

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

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

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

Â Because, notice, any horizontal line through this function.

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

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

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

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

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

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

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

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

Â other. That's because they're inverse functions

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

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

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

Â all, evaluate exponential and logarithm expressions.

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

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

Â and logarithm functions. Particularly when you have

Â transformations. You'll learn to solve exponential and

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

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

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

Â application problems. Speaking of applications, what are the

Â applications of exponentials and logarithms?

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

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

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

Â growth. Radioactive decay is another popular

Â application of exponentials and logarithms.

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

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

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

Â But we are looking at the decay of a substance.

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

Â interest. Compound interest is an example of an

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

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

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

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

Â another application of exponential and logarithms.

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

Â with this. The Richter scale for earthquakes is

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

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

Â Sound intensity is another thing that is measured in logarithms.

Â The sound intensity decibel levels, is a logarithmic funciton.

Â Function. And finally the learning curve, which is

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

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

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

Â actually will be modeled by exponentials and logarithms.

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

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

Â