EXPONENTIAL AND LOGARITHMIC FUNCTIONS (Part-1)

─►Exponential Functions

There is a function that has an important role not only in mathematics, but also in business, economics, and others areas of study. It involves a constant raised to a variable power, such as f(x) = a^x . We call such functions exponential functions.

The function f defined by

f(X)=a ^x 

 Where a > 0, a ≠ 1, and the exponent x is any real number, is called an exponential function with base a.

example :
Functions of the form:
f(x) = a^x
are known as exponential functions. The graphs of all such exponential functions pass through (0, 1).

Rules for Exponents

1. a^maⁿ = a^m+n
2. a^m/aⁿ = a^m-n
3. (a^m)ⁿ = a^mn
4. (a/b)ⁿ = aⁿ/bⁿ
5.  ^{a1 = a}
6. a⁰ = 1
7. (ab)ⁿ = aⁿbⁿ
8. ^{a-n = 1/an} 

 

Graphing Exponential Functions

As an experimental scientist you would start with the data, as we did in the examples of cancer incidence and viral decay, determine that the trend is exponential, and then find the best mathematical model to fit your data. However, it is often easier to understand different models if we consider the idealized curves. To better understand the features of exponential growth and decay, we will consider some general exponential functions through a mathematicians eyes.

Consider the following two exponential functions:

Let's construct a table of values for f₁ (x) and f₂ (x) to use as a guide when plotting.

x	- 4	-3	-2	-1	0	1	2	3	4
	16	8	4	2	1	1/2	1/4	1/8	1/16
	1/16	1/8	1/4	1/2	1	2	4	8	16

We can then plot the points listed in table above and draw a smooth curve through all of them.

Let's discuss some of the features of these graphical representations.
Exponential Decay 0 < a < 1 Properties As x → ∞,  f1 (x) → 0 This means the curve decrease as you look to the right because   f1 (x) approaches the y = 0 (properly called the horizontal asymptote). As x → - ∞,   f1 (x) → ∞ In other words the curve increases without bound to the left. If x = 0, f1 (x) = 1 The curve intersects the y-axis at the point (0, 1). This point is called the y intercept.	Exponential Growth a > 1 Properties As x → ∞,   f2 (x) → ∞ An exponential growth curve increases without bound as you look to the right because the value of f2 (x) increases as x increases. As x → - ∞,  f2 (x) → 0 The curve decreases as you look to left because the value of   f2 (x) approaches the horizontal asymptote, y = 0. If x = 0, f1 (x) = 1 The curve intersects the y-axis at the point (0, 1). This point is called the y intercept.

Notice that  f₁ (x) and  f₂ (x) exhibit contradictory behavior. In general any exponential function, y = a ^x, will look similar to either  

    or     ,

depending on whether a is less than or greater than one respectively.