DrDelMathCollege Algebra
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The following picture is essential to understanding composition of functions.
Composition of Absolute Value with Quadratic
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
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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. |
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| The most important of these infinitely many exponential functions is the one whose base is the irrational number e. |
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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". |
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Probably because of its relation to base 10 logarithms, the exponential function with base 10 is historically most significant. |
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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.
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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 |
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.
| 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 |
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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. |
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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.
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.
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 4.5: Exponential Growth and Decay