In this video, we introduce and explore exponential functions, give an example of exponential explosion and come across another very important number in mathematics called e, Euler's Number. We'll rapidly expand our repertoire of functions and move from very familiar types of numbers and operations to something sophisticated and powerful. To start with, consider any real number x and any positive integer n. Then x to the n is just x multiplied by itself with n occurrences of x appearing. So x to the 1 is just x, x squared is just x times x, x cubed is just x times x times x and so on. So if x is some particular number say 2, then 2 to the 1 is 2, 2 squared is 4, 2 cubed is 8 and so on. But we also have powers using zero and negative integers. It turns out to be very convenient to define x to the 0 to be 1 and x to the minus n to be 1 over x to the n. For example, 2 to the 0 is 1, 2 to the minus 1 is 1 over 2 to the 1 which is one-half, 2 to the minus 2 is 1 over 2 squared which is one quarter, 2 to the minus 3 is 1 over 2 cubed which is one-eighth and so on. We have the following exponential laws. The first says, "That to multiply two powers together, just add the exponents." The second says, "That to take a power of a power, just multiply the exponents." The third says, "That to divide one power by another, just subtract the second exponent from the first." The fourth says, "That to multiply two powers with different bases but the same exponent, just multiply the bases together." They should become familiar with practice and are very easy to verify when the exponents are integers. For example, 2 squared times 2 cubed is just 2 to the fifth because two plus three occurrences is 5 occurrences of 2 and you get 32 as predicted by the first exponential law. On the other hand, 2 squared divided by 2 cubed, and you can say directly simplifies to one-half which is 2 to the minus 1 and the exponent negative 1 is just 2 minus 3 as predicted by the third exponential law. If we take 2 squared and raise it to the third power, then we have three lots of 2 squared multiplied together, which is multiplying six lots of 2, yielding 64 as predicted by the second exponential law. If you multiply 2 squared by 3 squared, the exponent 2 is fixed and the base is changed from 2 to 3. We can rearrange this as two lots of 2 times 3 which is 6 squared giving 36 as predicted by the fourth exponential law. But we can go further and define fractional powers where the exponent is a fraction. Suppose that x is a positive real number and n is a positive integer, we define x to the 1 on n to be the positive nth root of x. That is the positive number which when raised to the nth power gives just x. Notice now the notation is consistent with one of our exponential laws since 1 on n multiplied by n is just 1 and x to the 1 is x. Now we can define fractional powers of x for any positive fraction. Let m and n be positive integers, So m on n is a positive fraction. To find x to the m on n to be either the nth root of x all raised to the nth power or alternatively, first form x to the m and then take the nth root. We end up at the same place consistent with the exponential laws since 1 on n times m and m times 1 on n are both just the fraction m on n. For example, 32 to the four-fifths may be taken to be the fifth root of 32, which is 2 all raised to the fourth power giving 16. We can always define a negative power by reciprocation. That is, x to the minus Alpha equals 1 over x to the Alpha for all exponents Alpha. We noted this before with positive integer exponents. This gives us in particular all negative fractional powers. For example, 32 to the minus three-fifths is the reciprocal of 32 to the three-fifths. We can rewrite this as 1 over 32 to the fifth all cubed, which becomes 1 over 2 cubed which is one-eighth. The exponential laws in fact hold for all positive real numbers used as bases and any exponents and you'll get used to them in no time. For example, we calculated 32 to the negative three-fifths before using the definitions. Alternatively, you could use one of the exponential laws to write this as 32 to the fifth all to the negative 3. So this becomes 2 to the negative 3, the reciprocal of 2 cubed which is one-eighth as before. Another example, take 81 to the three-quarters. You can rewrite this as the fourth root of 81 all cubed. But 81 is 9 squared, so you get these nested powers, 9 squared all to the quarter all cubed, and use an exponential law to simplify this to get 9 to the one-half all cubed. But the square root of 9 is 3, so you get 3 cubed which is 27. Here's some practice that looks tricky and your first reaction might be to use a calculator, probably expecting messy numbers. In fact, it simplifies quickly and cleanly using exponential laws. The numerator can be rewritten as 2 times 3 all to the 2.5 using one of the laws. 2.5 is just 5 on 2, 2 times 3 equals 6, in root 6 is just 6 to the half. And then this can all be rewritten as 6 to the 5 on 2 times 6 to the negative a half. Which becomes 6 to the 5 on 2 plus negative a half, which is 6 to the 4 on 2, which is 6 squared, which is 36. We now work towards introducing exponential functions, but first recall the shape of the rule for a power function. We have a variable base typically called x, but the exponent is fixed throughout. Now we change our point of view completely. Remember the power of lateral thinking to create what we call exponential functions. We now consider the base as fixed, a say, and the exponent as a variable. And again we'd like to use x as the variable. For the theory and apparatus to work in general, we need the base a to be positive. If you want a negative base, you need to work with complex numbers which is beyond the scope of this course. Here are some common examples and note the variable x is the exponent, not the base. y equals 2 to the x, 3 to the x, 0.5 to the x, 10 to the x and something called e to the x and I'll come back to that last one in a moment. 0.5 to the x is just a half to the x which is really just 1 over 2 to the x and that can be rewritten as 2 to the minus x. This last one e to the x uses perhaps the most important base in mathematics. The number e, also called Euler's number, and its decimal expansion begins 2.718. e is another famous and important rule number with nonrecurring decimal expansion. So, e is irrational and the proof of that is difficult. I'll explain it's importance in a moment. Euler was an 18th century mathematician. To have a number named after you is a great honor and the ultimate honor is to have a number called by the first letter of your surname in lowercase. Let's first try to understand and visualize the exponential function, y equals 2 to the x. We have a pair of axes and some values for a few inputs x, 2 to the 1 is 2, 2 squared is 4 and so on. 2 to the 0 is 1, 2 to the minus 1 is a half, 2 to the minus 2 is a quarter and so on. We can plot corresponding points on the xy-plane, and then join these points to get the full curve. It gets very steep very quickly as you move to the right. Let's try to think about how steep it gets. On the x-axis as we've drawn it here, the scale is about one unit per centimeter and on the y-axis, the scale has been made about four units per centimeter to fit in just these points. Now imagine if we wanted to include points on the graph for x equals six, seven, eight, nine and ten. Just an extra five centimeters of x axis. Clearly the y values are large and points to be plotted will go beyond this page. How long would the y-axis need to be to capture them all on the graph? With this scale on the vertical axis to accommodate an extra five centimeters horizontally, when you do the calculation, you would need a page which is about two and one-half meters long. This is an example of something called exponential explosion, often associated with population dynamics. I'll have more to say about this in a later video. Now what about Euler's number e, what's so special about it? I'll try to explain. First, draw the x-y axes and we'll keep the same scale on both axes. Then we add the line y equals x plus 1, which has slope 1 and passes through the y axis at y equals 1. Note that 1 is e to the 0, so this y intercept is also where the curve y equals e to the x crosses the y-axis. Now we have the curve of y equals e to the x and something nice happens. The line y equals x plus 1 becomes a tangent line to the curve at the y intercept. Euler's number e is chosen to shape or sculpture the exponential curve precisely to have this property and it's so important that we don't care that e happens to have a difficult decimal expansion. What this means is that the curve y equals e to the x as well as the tangent line has slope equal to one at the y intercept. The notion of slope of a curve will be formalized when we learn calculus in the next module. So, what? Well remember we mentioned in an earlier video, that self reference almost always leads to something useful in mathematics. In this case, we really do mean useful big time. A consequence of this slope and tangent property is that the function y equals e to the x has a self replicating property. After we define derivatives in the next module, you'll discover that the derivative of e to the x is just itself, e to the x. This is one of the most important facts in calculus and the key to developing a theory of differential equations. We've done a lot today. Discussed some lateral thinking, changing your point of view, applied to power functions where the variable is the base leads to exponential functions where the variable now is the exponent, explored an example of exponential explosion and introduced Euler's number e which has the property that the slope of the tangent to the curve, y equals e to the x to the y intercept is equal to 1 which has important consequences that we'll explore in future videos. Please read the notes and when you're ready, please attempt the exercises. Thank you very much for watching and I look forward to seeing you again soon.
