EXPONENTIAL AND LOGARITHMS FUNCTIONS (534)
2.1Exponential functions and their graph
An exponential function with base b is defined by the equation
[pic] Or [pic] ([pic] [pic]and x is a real number)
The domain of any exponential function is the interval ([pic] The range is the interval [pic]
Example 1:Graph the exponential function [pic]
Example 2: Graph the exponential function [pic]
Properties of exponential functions
1. The domain of any exponential function is the interval ([pic] 2. The range is the interval [pic]
3. The [pic]intercept is (0,1).
4. The [pic]axis is an asymptote of the graph.
5. The graph of [pic]passes through the point [pic]
2.2Logarithms functions and their graphs
Example 1:Graph the logarithms function [pic]
|x |f(x) |
|0.25 |-2 |
|0.5 |-1 |
|1 |0 |
|2 |1 |
|4 |2 |
Example 2:Graph the logarithms function [pic]
|x |f(x) |
|0.5 |2 |
|0.5 |1 |
|1 |0 |
|2 |-1 |
|4 |-2 |
An exponential function defined by y = b[pic], b > 0 and b [pic] 1, has an inverse function with the equation x = b[pic], b > 0 and b [pic] 1. To write the inverse function in the form y = f[pic](x), we need to take a look at the logarithms.
|Definition of logarithmic function: | |The logarithmic function with base b, b > 0 and b[pic] 1 is defined by y = log[pic] x if and only if x = b[pic]. |
Thus, the inverse of an exponential function y = b[pic] can be written in an exponential form (x = b[pic]) or a logarithmic form (y = log[pic] x). To translate from one to another, notice that x = b[pic] [pic] y = log[pic] x
Where y is the index and b is the base.
Example 1:Write the following in the exponential forms.
a.log[pic] 125 = 5
b.log[pic][pic] = -7
Example: 2Write the following in the logarithmic forms.
a.2[pic] = 5
b. 4[pic]= 3
2.3.1Common Logarithms and Natural Logarithms
There are only two bases of logarithms that can be readily obtained from mathematical tables or calculators. The first is a base of 10, known as common logarithms; the second is a base of e, called natural logarithms or Napierian logarithms. A common logarithm is denoted as log[pic]x or log x or just lg x. On the other hand, a natural logarithm can be written as log[pic]x or ln x.
2.3.2The Change-of-base Formula
To change the base of a logarithm log[pic]x to a new base a, we need to divide the base-a logarithm of x by the base-a logarithm of b.
Algebraically, the change-of-base formula states that if a, b and x are positive numbers, and a [pic] 1, b [pic] 1, then log[pic]x = [pic].
With the change-of-base formula, we can continue to evaluate log[pic]3 by changing to a base of 10. That is, log[pic]3 = [pic]
The complete solution for this question is presented as follow. 2[pic] = 3
log[pic]3 = x
x = [pic]
2.4 Properties of logarithms
Since logarithms are exponents, the basic rules of exponents have counterparts in logarithms. The basic rules of logarithms are as follow.
If x, y and b are positive numbers and b [pic] 1, then
1.log[pic]xy = log[pic]x + log[pic]y
2.log[pic][pic] = log[pic]x - log[pic]y
3.log[pic]x[pic] = r log[pic]x , r [pic] R
4.log[pic]1 = 0 (Because b[pic] = 1)
5. log[pic]b = 1 (Because b[pic] = b)...