- Exponential or Power Functions A Bit about e, the base of the Natural Logs
- The Exponential Function

- y = b
^{x}, raise the number b to the x power. - opposite function: y = - b
^{x} - reciprocal function: y = b
^{- x} - inverse function: y = log
_{b}x - slope: y = b
^{x}(ln b) - ex. y = 2
^{x}

The doubling function, for example, is a power function.

Use the rule: y = 2^{x}. Use values of 1, 2, 3, and 1/2 for x and
get corresponding values of 2, 4, 8, and
2 for y values.

Examine the "Family of Exponentials Graphs" to see how different
y = 1^{x} behaves relative to y = 1.5^{x}, y = 2^{x}, y = e^{x}.
For every value of x, 1^{x} is 1, so the function is
constant, there is no growth.

For all bases greater than 1, there is ALWAYS GROWTH, as x increases.

For all bases between 0 and 1, there is ALWAYS DECAY as x increases.

For positive values of x, the larger the base, 1.5 versus 2, versus e, the
larger the value of the function, y = 1.5^{x}, y = 2^{x}, y = e^{x}, and
therefore the steeper the curve, the faster the growth: The larger the base, the faster the growth.

For negative values of x, the smaller the base, 1.5 versus 2, versus e, the
larger the value of the function, y = 1.5^{x}, y = 2^{x}, y = e^{x}, the
and the steeper the curve, and the faster the growth: For negative values of x, the smaller the base, the
faster the growth.

The
slope of y = b^{x}
is b^{x}(ln b).

The
inverse function for
y = b^{x} is y = log_{b} x.

Power or exponential functions are powerful tools. They are used to express growth and decay. They are used to express other function such as:

- the normal distribution (the
bell curve
in statistics, a dilation and translation of e
^{-x²/2}) - hyperbolic functions such as

the hyperbolic sine, sinh x, is (e^{x}- e^{-x})/2,

the hyperbolic cosine, cosh x, is (e^{x}+ e^{-x})/2 - the logistic curve.

The numbers 0, 1, -1, , e, and i are celebrates among numbers. They are no different than other more common numbers, but are "more famous" and very useful and important. The numbers , 0, 1, and -1 are probably familiar to you. The complex number i is the square root of negative one, -1. It solves the equation x² = -1 with solutions of i and -i.

The number e is about equal to 2.71828 but this does not explain what number it is. It also may be said to solve an equation. One definition of e involves the solution of a calculus equation involving an integration, an antiderivative, the inverse of finding the slope. Here's the equation:

- b
- (1/x) dx = 1: The integral of the reciprocal from 1 to some upper limit equals one.
- 1

Consider the graph of the horizontal line y = 0 as a bottom boundary line. Use the vertical line x = 1 as a left boundary line, and the reciprocal function as a top boundary line. A four-sided shape would be created if a vertical line were drawn on the right. The vertical line which creates a shape having an area of one is the vertical line x = e. Suppose the integral of the reciprocal from 1 to some upper limit equals one. That upper limit is e. The number e solves the equation below where the upper limit b is the value being sought.

You may experiment with this yourself. Find the upper limit, the value that
makes the equation equal to 1.