EXPONENTIAL AND LOGARITHMS FUNCTIONS (534)

2.1Exponential functions and their graph

Definition

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 |

2.3Logarithms

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

Exercise

1. [pic]

2. [pic]

3. [pic]

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)...