7.4* General logarithmic and exponential functions

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1 7.4* General logarithmic and exponential functions Mark Woodard Furman U Fall 2010 Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

2 Outline 1 General exponential functions 2 The derivative 3 The logarithm to the base a. 4 Solving equations 5 e as a limit Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

3 General exponential functions Definition Let a > 0 be a real number (this will be the base). Notice that if r is rational, then a r = exp ( ln(a r ) ) = exp ( r ln(a) ). It makes sense, then, to define for all x R. a x = exp ( x ln(a) ). Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

4 General exponential functions The laws of exponents The laws of exponents are inherited from exp: a x+y = a x a y a x y = ax a y (a x ) y = a xy. In addition, we have the following two identities: (ab) x = a x b x (a/b) x = a x /b x. Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

5 General exponential functions The laws of exponents The laws of exponents are inherited from exp: a x+y = a x a y a x y = ax a y (a x ) y = a xy. In addition, we have the following two identities: (ab) x = a x b x (a/b) x = a x /b x. Graphs The graph of y = a x depends on whether a = 1, a > 1 or 0 < a < 1. Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

6 The derivative Theorem Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

7 The derivative Theorem d dx ax = a x ln(a) Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

8 The derivative Theorem d dx ax = a x ln(a) a x dx = ax ln(a) + C Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

9 The derivative Theorem d dx ax = a x ln(a) a x dx = ax ln(a) + C Find y : Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

10 The derivative Theorem d dx ax = a x ln(a) a x dx = ax ln(a) + C Find y : y = x2 x3 Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

11 The derivative Theorem d dx ax = a x ln(a) a x dx = ax ln(a) + C Find y : y = x2 x3 y = 2x 2 x 2 x + 2 x Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

12 The derivative Theorem d dx ax = a x ln(a) a x dx = ax ln(a) + C Find y : y = x2 x3 y = 2x 2 x 2 x + 2 x Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

13 The derivative Theorem d dx ax = a x ln(a) a x dx = ax ln(a) + C Find y : y = x2 x3 y = 2x 2 x 2 x + 2 x x 2 3 x3 dx Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

14 The derivative Theorem d dx ax = a x ln(a) a x dx = ax ln(a) + C Find y : y = x2 x3 y = 2x 2 x 2 x + 2 x x 2 3 x3 dx 1 dx x Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

15 The logarithm to the base a. Definition For a > 0, we will use log a to denote the inverse of a x. Thus log a (x) = y iff a y = x. Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

16 The logarithm to the base a. Definition For a > 0, we will use log a to denote the inverse of a x. Thus log a (x) = y iff a y = x. Theorem (Change of base formula) log a (x) = ln(x) ln(a) Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

17 The logarithm to the base a. Definition For a > 0, we will use log a to denote the inverse of a x. Thus log a (x) = y iff a y = x. Theorem (Change of base formula) log a (x) = ln(x) ln(a) Theorem d dx log a(x) = 1 x ln(a) Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

18 The logarithm to the base a. Find y for each of the following: Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

19 The logarithm to the base a. Find y for each of the following: y = log 10 (x 2 + 3x + 5) Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

20 The logarithm to the base a. Find y for each of the following: y = log 10 (x 2 + 3x + 5) y = x x Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

21 Solving equations In each case, solve for x: Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

22 Solving equations In each case, solve for x: 2 x = 2/8 Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

23 Solving equations In each case, solve for x: 2 x = 2/8 2 x 35 2 x = 2 Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

24 e as a limit Theorem (Hint: What is d dx ln x x=1?) ln(1 + h) lim = 1 h 0 h Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

25 e as a limit Theorem (Hint: What is d dx ln x x=1?) Corollary ln(1 + h) lim = 1 h 0 h lim (1 + h 0 h)1/h = e Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

26 e as a limit Theorem (Hint: What is d dx ln x x=1?) Corollary ln(1 + h) lim = 1 h 0 h lim (1 + h 0 h)1/h = e Evaluate ( lim ) n n n Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall / 9

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