Capter 3 Review

Chapter 3 Review
Chapter 3 is about Logarithmic and Exponential Functions

3.1-Exponential Functions and Their Graphs

A basic Exponential Function is written as  
Where a>0  and a doesn't equal 1.
It's graph is as shown. It is an infinite domain, but a range of only 0 to infinity.

The graph of  can be transformed in 4 different ways. To illustrate this can be used.
Where a = vertical stretch of the function
           b = a sharper turn for higher b and a shallower turn for lower b
           c = horizontal shift of the function in the inverse direction of b's sign
           d = vertical shift of the function

3.2-Logarithmic Functions and Their Graphs

A basic Logarithmic Function is written as
Where a>0, x>0 and a doesn't equal 1.
Also, if and only if  does .
It is the inverse function of the basic exponential function and looks like this. Its range is negative infinity to 1 and its domain is 0 to infinity.


Common Log:
Natural Log:

The graph of can be transformed in 4 different ways. To illustrate thiscan be used.
Where a = vertical stretch of the function
           b = a sharper turn for higher b and a shallower turn for lower b
           c = horizontal shift of the function in the inverse direction of b's sign
           d = vertical shift of the function

 

3.3-Properties of Logarithms

Change of Base Formula:

If a, b, and x are positive real numbers and a and b don't equal 1. Then

This works for any base that is convenient as long as the bottom of the fraction is taking some sort of log of the initial base.

Properties of Logarithms:

Let a be a positive real number that doesn't equal 1, and let n be a real number as well. If u and v are positive real numbers, the following properties are true.

3.4-Solving Exponential and Logarithmic Equations

There are two basic strategies to solving exponential and logarithmic equations, the one-to-one property and the inverse properties.

One-to-One Properties:
 if and only if x=y.
if and only if x=y.
This property states that there is only one x value for every y value and therefore the log of a number or something taken to a number's power must equal other logs with the same base and other exponents with the same base.

Inverse  Properties:

This property states that a log taken of the base number is equal to the base number's power. This works along the same lines of 2/(2*4) because logs are the inverse operation of exponents just like division is the inverse operation of multiplication.


To solve a Logarithmic Equation in the easiest way, first write it in exponential form. Then use one-to-one or inverse properties as necessary to simplify the equation.

3.5-Exponential and Logarithmic Models
Mathematical models are equations and applications of math that are used to figure out parts of the real world. There are five common types involving exponential and logarithmic functions.

1. Exponential Growth:


2. Exponential Decay:

3. Gaussian Model:


4. Logistic Growth:


5. Logarithmic: