1. The Exponential Function
Integer Powers
Fractional Powers and Irrational Powers
Properties of Powers
The Mathematical Constant e
The Exponential Function

2. Functions/Elementary Functions/The Exponential Function by M. Seppälä
General Powers?
We all know that
22 = 2  2 = 4.
Starting Question
What means
2 ?
Functions/Elementary Functions/The Exponential Function by M. Seppälä

3. General Power Functions
For a number a, one defines the positive integer powers of a recursively by setting
a1 = a,
a2= a  a,
an+1 = a  an for n > 1.
Negative integer powers are defined by setting for n = 1, 2, ….
Assuming that a ≠ 0, one defines a0 by setting a0 = 1.
Warning
00 is undefined. One cannot assign a value to 00. This is an example of indeterminates. In computer mathematics systems such expressions are called NaNs, short for “Not a Number.”
Functions/Elementary Functions/The Exponential Function by M. Seppälä

4. Functions/Elementary Functions/The Exponential Function by M. Seppälä
Roots
Let n be a positive integer, and a > 0. Let b be a positive number such that bn = a. Requiring b be positive makes it uniquely defined.
Definition
The number b is the positive nth root of the positive number a.
Notation
Warning
Even roots of negative numbers are not real numbers, i.e., there are no real numbers whose even power would be negative.
If n is a positive even number and a positive, the equation bn = a has always two solutions b and -b. If n is odd, the solution is unique and positive.
Functions/Elementary Functions/The Exponential Function by M. Seppälä

5. Fractional and Irrational Powers
Definition
Let p be an integer, and q a positive integer. For a positive number a define the fractional power ap/q by setting
The precise definition of aρ for irrational numbers ρ can be given by approximating an irrational number ρ by rational numbers p/q, and then approximating the irrational power aρ by the rational powers ap/q. In this way ax can be defined for all real numbers x assuming that a > 0.
We will not go into more details at this point. These considerations can be made rigorous. The general power can also be defined using integration. We will later use this method to give a precise definition of general powers.
Functions/Elementary Functions/The Exponential Function by M. Seppälä

6. Functions/Elementary Functions/The Exponential Function by M. Seppälä
Properties of Powers
The following properties of powers follow from the definition. Here we assume that a is positive. 1 2 3 4
If a > 1, x > y  ax > ay. 5
If 0 < a < 1, x > y  ax < ay. 6
Functions/Elementary Functions/The Exponential Function by M. Seppälä

7. Exponential Functions
General exponential functions are functions of the form f(x) = ax for some positive number a.
The figure on the right shows the graphs of the functions
y = (1/2)x, the red curve,
y = 1x, the black line,
y = (3/2)x, the blue curve, and
y = (5/2)x, the green curve.
Observe that, if a > 1, the function y = ax is increasing. The greater the number a is, the faster the values of the function y = ax increase. If 0 < a < 1, the function y = ax is decreasing.
If a ≠ 1 and a > 0, the function y = ax is monotonic bijection between real numbers and positive real numbers, y:  +, y = ax.
Functions/Elementary Functions/The Exponential Function by M. Seppälä

8. Functions/Elementary Functions/The Exponential Function by M. Seppälä
The Number e
a=1/2
a=1
a=3/2
a=5/2
From the picture it is obvious that, as the parameter a grows, the slope of the line, tangent to the graph of the function ax at the point (0,1), grows.
Definition
The mathematical constant e is defined as the unique number for which the slope of the tangent line of the graph of the function ex at x = 0 is 1.
The slope of a tangent line is the tangent of the angle at which the tangent line intersects the x-axis.
e 
Functions/Elementary Functions/The Exponential Function by M. Seppälä

9. The Exponential Function
Definition
The function y = ex is the exponential function.
Since the mathematical constant e is greater than 1, the exponential function is an increasing bijection between the set of real numbers and the set of positive real numbers.
The exponential function is an important basic function in calculus. Here we have defined the basic mathematical constant e using a rather heuristic geometric condition. The constant e can be defined more rigorously by consider the expression
for large values of the integer n. One can show that, as n grows, the numbers (1 + 1/n)n approach the mathematical constant e. We will make these remarks precise later.
Functions/Elementary Functions/The Exponential Function by M. Seppälä