Numerical Methods. Exponential and Logarithmic functions. Jaesung Lee

Transcription

1 Numerical Methods Exponential and Logarithmic functions Jaesung Lee

2 Exponential Function

3 Exponential Function Introduction We consider how the expression is defined when is a positive number and is irrational. We consider these exponential functions ( ) = in more depth and in particular consider the special case when the base a is the socalled exponential constant: = We then examine the behavior of as called exponential growth and of as called exponential decay. 3

4 Exponential Function Exponents To represent a series of multiplication of the same number, known as factor, can be used where is the base and is called the exponent. There are various cases to consider: = with factors / means the root of. That is, / is that positive number which satisfies: / / / = 4

5 Exponential Function Exponents / means the root of. That is, / is that positive number which satisfies: / / / = where there are factors on the left hand side. / = / / where there are factors. =. 5

6 Exponential Function Exponents For convenience, we list the basic laws of exponents: = = = = = 1 where 0 6

7 Exponential Function What is if is a real number? So far we have given the meaning of where is, at worst, a rational number, that is, one which can be written as a quotient of integers. So, if is rational, then = where, are integers Now consider as a real number which cannot be written as a rational number. Two common examples of these irrational numbers are = 2 and =. 7

8 Exponential Function What is if is a real number? What we shall do is approximate by a rational by working to a fixed number of decimal places. For example if = then, if we are working to 3 decimal places, we would write and this number can be expressed as a rational number: = 8

9 Exponential Function What is if is a real number? So, in this case =.. = and the final term: can be determined in the usual way. From henceforth, we shall therefore assume that the expression is defined for all positive values of and for all values of. 9

10 Exponential Function Details of exponential functions For a fixed value of the base, the expression varies with the value of : it is a function of. Below graphs show 0.5, 0.3, 1, 2 and 3. 10

11 Exponential Function Details of exponential functions The functions (as different values are chosen for ) are called exponential functions. From the graphs, we see: If > > 0 then > if > 0 and < if < 0 These are true for all exponential functions. 11

12 Exponential Function Details of exponential functions The most important and widely used exponential function has the particular base , a number always denoted by the single letter : = Note that cannot be written as the quotient of two integers. The exact definition of can be written as: = lim which eventually leads to a precise value of. 12

13 Exponential Function Exponential growth If > 1 then it can be shown that, no matter how large is: as That is, if is fixed then as increases, will overtake the value as long as > 1. The behavior of as is called exponential growth. 13

14 Exponential Function Exponential decay In a similar manner, we characterize the behavior of the function as as exponential decay. The graphs of and are shown in the above diagram. In fact, tends to zero so quickly as that, no matter how large the fixed number is, 0 as 14

15 Logarithms

16 Logarithms Introduction In this section, we introduce the logarithm: log. The operation of taking a logarithm essentially reverses the operation of raising a base to a power. We will formulate the basic laws satisfied by all logarithms and learn how to manipulate expressions involving logarithms. 16

17 Logarithms Introduction Logarithms are introduced to reverse the process of raising a base to a power. As with all exponentials we demand that the base should be a positive number. The expression log = is read the log to base of the number is equal to Key point If = then log =. If log = then =. 17

18 Logarithms Some exercises Find the log equivalents of = Based on the key point, the log equivalents of = is log =. = Based on the key point, the log equivalents of = is log =. = = Based on the key point, the log equivalents of = is log = +. 18

19 Logarithms Some exercises From the last guided exercise we have found, using the property of indices, that log = + = log + log We conclude that the index law = has an equivalent logarithm law log = log + log in words the log of a product is the sum of logs This is one of the major advantages of using logarithms. 19

20 Logarithms The laws of logarithms log = log + log log = log log log 1 = 0, log = 1 log = log 1 = log + log = log log log = log times times log = log = log + + log = log 20

21 Logarithms Simplifying expressions involving logarithms To simplify an expression involving logarithms, their laws need to be used. For example, log 2 log 4 + log 4 + log 3 4 can be simplified as log = log 2 3 = log 2 + log 3 = log

22 Logarithms Changing bases in logarithms It is sometimes required to express the logarithm w.r.t. one base in terms of a logarithm w.r.t. another base. Now = implies log = where we have used logs to base. What happens if we want to use another base? 22

23 Logarithms Changing bases in logarithms We take logs to base of both side of = : log = log = log By dividing log to both sides, we have: = that is log = 23

24 Logarithms Another important result Simplify the expression 10. Let = 10 then take logs to base 10 of both sides: log = log 10 = log log 10 where we have used: log = log. 24

25 Logarithms Another important result Simplify the expression 10. log = log 10 = log log 10 Since we are using logs to base 10, it can be simplified as log = log Because log = log implies =, 10 = 25

26 Logarithms Another important result Simplify the expression = This is an important results and can be generalized to logarithms of other bases: = indicating that raising to the power and taking logs are inverse operations. 26

27 Logarithmic Function

28 The Logarithmic Function Introduction In this section, we consider the logarithmic function = log and examine its important characteristics. We see that this function is only defined if is a positive number. We also see that the log function is the inverse function of the exponential function and vice versa. We show, through numerous examples, how equations involving logarithms and exponentials can often be solved. 28

29 The Logarithmic Function Introduction We introduced the operation of taking logarithms which essentially reverses the operation of exponentiation. If > 0 and 1 then = implies = log In this section, we consider the function log in more detail. We shall concentrate only on the functions log (i.e. to base 10) and ln (i.e. to base ). 29

30 The Logarithmic Function Introduction In any case the logarithmic function to any other base can be re-written in terms of log or ln since: log = and also log = Both of the functions = log and = ln have similar characteristics. We can never choose as a negative number since 10 and are each always positive. 30

31 The Logarithmic Function Illustration of the logarithmic function The graphs of = log and = ln are shown in the following diagram. 31

32 The Logarithmic Function Inverse of the logarithmic function From the graphs we see that both functions are one-to-one so each has an inverse function. In fact, as is easy to demonstrate, the inverse function of log is. Let us do this for logs to base 10. Let = log then is, by definition, precisely that function which takes log as the input to produce an output. 32

33 The Logarithmic Function Inverse of the logarithmic function We claim that ( ) = 10. To see this, we replace by log (as the input) then log = 10 To simplify let = 10 and take logs of both sides. log = log 10 = log log 10 = log so = That is = 10. This result proves that the inverse function to log is 10. In a similar way the inverse function to ln is. 33

34 The Logarithmic Function Solving equations involving logarithms and exponentials 3 = 10 = 4 = = 0 34