DrDelMath College Algebra by Stewart/Redlin/Watson |
Chapter 5: Exponential and Logarithmic Functions
Section 5.1: Exponential Functions
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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.
Section 5.2: Logarithmic Functions and Their Graphs
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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 |
Section 5.3: 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 |
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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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Section 5.4: 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.
Section 5.5: Exponential and Logarithmic Models