DrDelMath

SUMMARY
Chapter 5:  Exponential and Logarithmic Functions

Section 5.1: Exponential Functions and Their Graphs

Goals -- What You Should Learn

1. The definition of the exponential function base a
2. The definition of the exponential function base e
3. Characteristics of the graphs of exponential functions
4. Recognize the graph of exponential functions
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- Vocabulary -
Definitions to Memorize
Always review previously memorized definitions

1.  Exponential function base 2
2.  Exponential function base e
3. Exponential function base 10
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Links to Supplemental Material
Check OnLine Drill and Practice

1. Manipulating Exponentials, Radicals, and Logarithms

Comment: An important irrational number is named e. The first few digits in a decimal expression of e is given here:
        
Other irrational numbers encountered earlier have had understandable explanations. It is easy to understand what is meant by the square root of 2. It may not be clear why it is irrational, but at least its meaning is clear. In the same way the number is easily understood to be the ratio of the circumference of a circle divided by its diameter. The number e on the other hand has no simple explanation.

You should review the rules for manipulating exponentials and logarithms as summarized in Special Topics.

Definition: Suppose a is a positive real number. The base a exponential function is named expa and its rule is expa(x)= ax. The domain of expa is all real numbers.

Graphs of several exponential functions are show at the right:

 

Definition: The base 2 exponential function is named exp2 and its rule is exp2(x)= 2x. The domain of exp2 is all real numbers.
Definition: The base 10 exponential function is named exp10 and its rule is exp10(x)= 10x. The domain of exp10 is all real numbers.
Definition: The base e exponential function is named expe and its rule is expe(x)= ex. The domain of expe is all real numbers.

Note that for each exponential function
    1) The y-intercept is 1
    2) There are no x-intercepts -- hence no real zeros
    3) The graph is entirely above the x-axis -- that means expa(x) > 0 for all real numbers x
    4) No horizontal line intersects a graph in more than one point -- hence each exponential function has an inverse function -- the corresponding logaritmic function.  
    5) The x-axis is a horizontal asymptote for each exponential function.

Comment: The most important of these exponential functions is the base e exponential function. Most of our work with exponential functions and their inverse functions will be with the base e exponential and logarithm.

 

Section 5.2: Logarithmic Functions and Their Graphs

Goals -- What You Should Learn

1. The definition of the logarithmic function base a
2. The definition of the logarithmic function base e
3. Characteristics of the graphs of logarithmic functions
4. Recognize the graph of logarithmic functions.
5.
6.
7.

- Vocabulary -
Definitions to Memorize
Always review previously memorized definitions

1.  Logarithmic function base 2
2.  Logarithmic function base e
3. Logarithmic function base 10
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6.  
Links to Supplemental Material
Check OnLine Drill and Practice

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Definition: The base 2 logarithmic function is named log2 and is defined to be the inverse of exp2. The domain of log2 is all positive real numbers.  
Definition: The base 10 logarithmic function is named log and is defined to be the inverse of exp10. The domain of log is all positive real numbers.
Definition: The base e logarithmic function is named ln and is defined to be the inverse of exp. The domain of ln is all positive real numbers.

Note that for each logarithmic function
    1) There is no y-intercept
    2) The x-intercept is 1 -- hence each has a real zero equal to 1
    3) The graph is entirely to the right of the y-axis -- that means the domain of these functions is all positive real numbers
    4) No horizontal line intersects a graph in more than one point -- hence each exponential function has an inverse function -- the corresponding exponential function.
    5) The y-axis is a vertical asymptote for each logarithmic function.

Comment: The most important of these logarithmic functions is the base e exponential function named ln. Most of our work with logarithmic functions will be with the base e exponential and logarithm.

Properties:  
ln(1) = 0 e0 = 1
ln(e) = 1 e1 = e
ln(ex) = x eln(x) = x
If ln(x) = ln(y) then x = y.  

 

Section 5.3: Properties of Logarithms

Goals -- What You Should Learn

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- Vocabulary -
Definitions to Memorize
Always review previously memorized definitions

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6.  
Links to Supplemental Material
Check OnLine Drill and Practice

1. Manipulating Exponentials, Radicals, and Logarithms

The following Change of Base formulas provide a means for calculating functional values for logarithmic functions regardles of the base. In reality, base 10 and base e are about the only ones used and base e or the ln function is most prevalent.

                                  

All the important properties of logarithmic functions are presented in the Supplemental Material.

Section 5.4: Exponential and Logarithmic Equations

Goals -- What You Should Learn

1. To use the fact that ln and exp are inverses of each other
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- Vocabulary -
Definitions to Memorize
Always review previously memorized definitions

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6.  
Links to Supplemental Material
Check OnLine Drill and Practice

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Note: All previously learned techniques for solving equations are applicable to logarithmic and exponential equation.

Note: Logarithmic and exponential equations are solved by making use of the fact that the logarithm function and exponential function with the same base are inverses of each other.

Note: In particular, use the fact that ln and exp are inverses of each other to solve equation involving either ln or e.

Section 5.5: Exponential and Logarithmic Models

Goals -- What You Should Learn

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- Vocabulary -
Definitions to Memorize
Always review previously memorized definitions

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3. 
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5.  
6.  
Links to Supplemental Material
Check OnLine Drill and Practice

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