Logarithmic Functions and Models Power Functions Logistic Function. Mathematics. Rosella Castellano. Rome, University of Tor Vergata

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1 Mathematics Rome, University of Tor Vergata

2 The logarithm is used to model real-world phenomena in numerous elds: i.e physics, nance, economics, etc. From the equation 4 2 = 16 we see that the power to which we need to raise 4 in order to get 16 is 2. Another way of writing the equation 4 2 = 16 is log 4 16 = 2. Base b Logarithm The base b logarithm of x, log b x, is the power to which we need to raise b in order to get x. Symbolically: log b x = y means b y = x.

3 Example The following table lists some exponential equations and their equivalent logarithmic forms:

4 Common Logarithm log 10 x = logx Natural Logarithm log e x = lnx Examples Logarithmic Form Exponential Form 1. log10, 000 = = 10, ln e = 1 e 1 = e

5 In general, consider the exponential equation b x = a with a, b 2 < and x unknown. Given a > 0, b 6= 1, b > 0 it exists one and only one real number x such that b x = a. De nition x which solves b x = a is the logarithm in base b of a: log b a. It is basically a power! 2 5 = 32 log 2 32 = 5

6 log b a = log a log b = ln a ln b Example log 11 9 = log 9 log 11 = ln 9 ln

7 A logarithmic function has the form: f (x) = log b x + C, f (x) = log x or, alternatively, f (x) = A ln x + C. f (x) = ln x 5

8 The logarithm function with base b is the function y = log b x. with b > 0 and b 6= 1. The function is de ned for all x > 0.

9 Note the following: For any base, the x-intercept is 1; The graph passes through the point (b, 1); The graph is below the x-axis the logarithm is negative for 0 < x < 1; The function is de ned only for positive values of x.

10 Simple Logarithm Properties 1 log a xy = log a x + log a y 2 log x a y = log a x log a y 3 log a (x) k = k log a x 4 log a a = 1; log a 1 = 0 5 log a 1 x = loga x 6 log a x = log b x log b a

11 Example 1 log 4 28 = log log log = log2 15 log log 2 (7 15 ) = 15 log log 5 5 = 1; ln e = 1; log 11 1 = 0 5 log = log2 5 6 log 4 28 = log 5 28 log 5 4 = log 28 log 4

12 The following two identities demonstrate that the operations of taking the base b logarithm and raising b to a power are inverse to each other. The inverse of any exponential function is a logarithmic function for any base b: 1 log b (b x ) = x, log 2 (2 7 ) = 7 2 b log b x = x 5 log 5 8 = 8 Identity 2) embodies the de nition of a logarithm: log b x is the exponent to which b must be raised to produce x.

13 These identities satisfy the de nition of a pair of inverse functions. Therefore for any base b, the functions f (x) = b x and g(x) = log b x are inverses.

14 Example Global bonds sold by Mexico are yielding an average of 2.51% per year. At that interest rate, how long will it take a $1, 000 investment to be worth $1, 200 if the interest is compounded monthly?

15 Solution: Substituting FV = 1, 200, PV = 1, 000, r = , and m = 12 in the compound interest equation gives FV = PV 1 + r mt m 1, 200 = 1, t 12 t = ln ( )

16 An exponential decay function has the form Q(t) = Q 0 e kt, Q, k > 0 Q 0 represents the value of Q at time t = 0, and k is the decay constant. The decay constant k and half-life t h for Q are related by t h k = ln 2

17 Example If t h = 10 years, then 10k = ln 2, so k = ln 2 10 decay model is and the Q(t) = Q 0 e t

18 Exponential Growth Model and Doubling Time An exponential growth function has the form: Q(t) = Q 0 e kt Q 0 represents the value of Q at time t = 0, and k is the growth constant. The growth constant k and doubling time t d for Q are related by t d k = ln 2. Example P(t) = 1, 000e 0.05t

19 The following table shows the total spent on research and development by universities and colleges in the U.S., in billions of dollars, for the period (t is the number of years since 1990). Find the best- t logarithmic model of the form S(t) = A ln t + C and use it to project total spending on research by universities and colleges in 2012, assuming the trend continues.

20 We use technology to get the following regression model: S(t) = 19.3 ln t Because 2012 is represented by t = 22, we have S(22) = 19.3 ln(22)

21 A function of the form f (x) = x a where a is a constant is called a power function. a = n where n is a positive integer. The graphs of f (x) = x n can be obtained cosidering polynomials with only one terms: the general shape depends if n is even or odd. a = 1 n. the function f (x) = x 1 n = np x is a root function. For instance f (x) = p x for a = 1 we have the reciprocal function f (x) = 1 x

22 Power functions (graphs) a = n where n is a positive integer. n=1,2,3,4,5, y x 100

23 Power functions (graph) a = 1 n For instance f (x) = p x y x

24 Modeling with the Logistic Regression A logistic function has the form N f (x) = 1 + Ab x for nonzero constants N, A, and b (A and b positive and b 6= 1). Example N = 6, A = 2, b = 1.1 gives f (x) = (1.1 x ) the y intercept is N/(1 + A): f (0) = = 2 when x is large, f (x) N: 6 f (1, 000) = 1+2(1.1 1,000 ) = 6 = N

25 Modeling with the Logistic Regression

26 Modeling with the Logistic Regression Properties of the Logistic Curve The graph is an S shaped curve sandwiched between the horizontal lines y = 0 and y = N. N is called the limiting value of the logistic curve. If b > 1 the graph rises; if b < 1, the graph falls. The y-intercept is N 1+A The curve is steepest when t = ln A ln b

27 Modeling with the Logistic Regression for small x and the role of b For small values of x, we have: N N 1 + Ab x b x. 1 + A Thus, for small x, the logistic function grows approximately exponentially with base b.

28 Modeling with the Logistic Regression 50 Let f (x) = 1+24(3 x ) Then f (x) (3x ) = 2 (3 x ) for small values of x. The following gure compares their graphs: Figure: The blue curve is the exponential

29 Modeling with the Logistic Regression A u epidemic is spreading through the U.S. population. An estimated 150 million people are susceptible to this particular strain, and it is predicted that all susceptible people will eventually become infected. There are 10,000 people already infected, and the number is doubling every 2 weeks. Use a logistic function to model the number of people infected. Hence predict when, to the nearest week, 1 million people will be infected.

30 Modeling with the Logistic Regression Solution Let t be time in weeks, and let P(t) be the total number of people infected at time t. We want to express P as a logistic function of t, so that P (t) = N 1+Ab t. We are told that, in the long run, 150 million people will be infected, so that N = 150, 000, 000 (limiting value of P). At the current time (t = 0), 10, 000 people are infected, so 10, 000 = N 1+A = 150,000,000 1+A (value of P when t = 0).

31 Modeling with the Logistic Regression Solving for A gives 10, 000(1 + A) = 150, 000, A = 15, 000 A = 14, 999. What about b? At the beginning of the epidemic (t near 0), P is growing approximately exponentially, doubling every 2 weeks. We found that the exponential curve passing through the points (0, 10, 000) and (2, 20, 000) is: so b = p 2 p t y = 10, 000 2

32 Modeling with the Logistic Regression Now that we have the constants N, A, and b, we can write down the logistic model: The graph of this function is: 150, 000, 000 P (t) = p t

33 Modeling with the Logistic Regression Now we tackle the question of prediction: When will 1 million people be infected? In other words: when is P(t) = 1, 000, 000?

34 Modeling with the Logistic Regression Let us consider some data on the percentage of Internet-connected households with broadband and try to estimate the percentage of households that will have broadband in the long term. Since we require a model for the data, we need to do some form of regression. See Excel File Logistic Regr.