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DrDelMath

College Algebra
by Blitzer
SUMMARY

 

 

Chapter 4:   Exponential and Logarithmic Functions

Section 4.0: Background and Review

The following picture is essential to understanding composition of functions.

Picture of Composition of Functions

Section 4.1: Inverses of Functions

Function: Inverses

The following diagram illustrates the relationship between a function and its inverse.
Think of the red arrows as the rule for a function f and the blue arrows as the rule for its inverse f-1

Picture of Inverse Functions

Minimal List of Exercises Page 290.

If you understand the previous material you should be able to answer the following questions.
Each of these questions should be answered. Most of these questions are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.
Section 2.7 Exercise Set: 1,3,5,7,9,11,13,15,21,23,25,29-34.

 

Section 4.2: Exponential Functions



The number 1 is excluded from the possible bases because if a =1, the corresponding exponential function with 1 as its base is simply the constant function whose rule is c(x) = 1. It is preferable to not consider this constant function to be an exponential function.

Of course there are infinitely many such exponential functions. Graphs of a few of them are shown at the right.

graphs of exponential functions

 

The most important of these infinitely many exponential functions is the one whose base is the irrational number e. graph of exp

 

Another important one of these infinitely many exponential functions is the one whose base is the number 2. This function is important because it frequently models "exponential growth and decay".

graph of exp base 2

 

Probably because of its relation to base 10 logarithms, the exponential function with base 10 is historically most significant.

graph of exp base 10

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 its inverse the natural logarithm named ln.

Sample Exercises

Section 4.3: Logarithmic Functions

     
 

Comment: The most important of these logarithmic functions is the base e logarithm 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. If ex = ey, then x = y

 

Minimal List of Exercises Page 434.

If you understand the previous material you should be able to answer the following questions.
Each of these questions should be answered. Most of these questions are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.

Section 4.2 Exercise Set: 1 - 8, 9 - 20, 21 - 42, 75, 77, 87 - 98.

Section 4.4: Properties of Logarithms

Properties of the ln Function Corresponding Property of the exp Function
  Function Notation Traditional Notation
ln(1) = 0 exp(0) = 1 e0 = 1
ln(e) = 1 exp(1) = e e1 = e
ln(exp(x)) = x exp(ln(x)) = x ln(ex) = x

eln(x) = x
If ln(x) = ln(y),
then x = y
If exp(x) = exp(y),
then x = y
If ex = ey,
then x = y
ln(xy) = ln(x) + ln(y) exp(x)exp(y) = exp(x + y) exey = ex+y
ln(x/y) = ln(x) - ln(y) exp(x)/exp(y) = exp(x - y) ex/ey = ex-y
ln(xy) = yln(x) [exp(x)]y = exp(xy) (ex)y = exy
     

 

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.

Minimal List of Exercises Page 445.

If you understand the previous material you should be able to answer the following questions.
Each of these questions should be answered. Most of these questions are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.

Section 4.3 Exercise Set: 13, 15, 19, 20, 30, 37, 38, 43, 44, 51, 52, 55 - 62.

Section 4.5: Exponential and Logarithmic Equations

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

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 equations involving either ln or e.

Sample Exercises

 

Minimal List of Exercises Page 456

If you understand the previous material you should be able to answer the following questions.
Each of these questions should be answered. Most of these questions are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.

Section 4.4 Exercise Set: 25, 26, 29 - 36, 43 - 46, 51, 52, 59 - 64, 75 - 82, 89, 90.

Section 4.6: Exponential Growth and Decay

Section 4.7: Applications of Exponential and Logarithmic Function