1. Exponential and Logarithmic Functions Chapter 11

2. 11.1 Real Exponents Review properties of exponents

3. 11.1 Real Exponents Worksheet For Notes and Examples

4. 11.2 Exponential Functions Graph exponential functions and inequalities Solve problems involving exponential growth and decay

5. 11.2 Exponential Functions Power Function – Functions in which the variable is the base and the exponent is any real number – I.E. y = x 5 Exponential Function – Functions of the form y=b x in which the base b is a positive real number and the exponent is a variable – I.E. y = 5 x

6. 11.2 Exponential Functions Consider the graph for y = 2 x – In order to graph we have to plug in values for x and plot the points – Use graphing calculator

7. 11.2 Exponential Functions Exponential Decay – When a quantity loses value exponentially over time – Examples: Temperature of Coffee Exponential Growth – When a quantity increases value exponentially over time – Examples: Savings account balance

8. 10.1 Exponential Functions b tells you Growth vs. Decay Growth: a>0Decay: 0<a b>10<b<1 y-intercept is a Find on calc by: 2 nd calc – value – x=0

9. Are these growth or decay? Y=(1/5) – decay Y=3(4) x – Growth

10. 11.2 Exponential Functions Formula for Exponential Growth/Decay – N = n 0 (1+r) t N is the final amount n 0 is the initial amount r is the rate of growth or decay per time period t is the number of time periods.

11. 11.2 Exponential Functions Example: – The average growth rate of the population of a city is 7.5% per year. The city’s population is now 22,750 people. What is the expected population in 10 years? Set up Equation – N = N o (1.075) x – N = 22750(1.075) 10 In 10 years, the population will be about 46, 889 people

12. 11.2 Exponential Functions Compound Interest – A = P(1+r/n) nt P is the principle or initial investment A is the final amount of the investment r is the annual interest rate n is the number of times interest is paid t is the number of years

13. 11.2 Exponential Functions Determine the amount of money provided an annual rate of 5% compounded daily if Marcus invested $2000 and left it in the account for 7 years. – A = 2000*(1+0.05/365) 365*7 – A = $2838.06 after 7 years

14. 11.2 Exponential Functions How much should Sabrina invest now in a money market account if she wishes to have $9000 in the account at the end of 10 years? The account provides an APF of 6% compounded quarterly. – 9000=X*(1+0.06/4) 4*10 – X = $4961.36

15. Practice Problems An area had a population of 700,000 in 1990. The average yearly rate of growth is 5.9%. Find the projected population for 2010. Determine the amount of money in a savings account that provides an annual rate of 4% compounded month if the initial investment is $1000 and the money is left in the account for 5 years. How much money must be invested by Mr. Kaufman if he wants to have $20,000 in his account after 15 years? He can earn 5% compounded quarterly.

16. 11.3 The number e Use the exponential function y = e x

17. 11.3 The number e Exponential Growth or Decay in terms of e – N = N o e kt N is the final amount N o is the initial amount k is a constant t is time e is not a variable, but a number

18. 11.3 The number e A beaker of liquid cools exponentially when removed from a source of heat. Assume the the initial temperature is 90 degrees and that k = 0.275. – Function equation? T = T i e -kt – Find the temperature after 4 minutes About 30 degrees

19. 11.3 The number e Continuously Compounded Interest – A = Pe rt P is the initial amount A is the final amount R is the annual interest rate T is time in years

20. 11.3 The number e Compare the balance after 30 years of a $15,000 investment earning 12% interest compounded continuously to the same investment compounded quarterly. – Continuously A = Pe rt A = 15000*e.12*30 – Quarterly A = P(1+r/n) nt A = 15000(1+.12/4) 4*30 About 5.44% more

21. Section 11.4 Logarithmic Functions Evaluate Logarithms in expressions Solve equations and inqualities involving logarithms

22. 11.4 Logarithmic Functions Logarithmic Function – Y = log a x A >0 and a <>1 The inverse of the exponential function x=a y

23. 11.4 Logarithmic Functions Y = log a x X=a y Y is the exponent A is the base X is the solution

24. 11.4 Logarithmic Functions Write each equation in exponential form – 2/3 = log 125 25 25=125 2/3 – Log 27 3=1/3 3=27 1/3

25. 11.4 Logarithmic Functions Write each equation in logarithmic form – 4 3 =64 Log 4 64=3 – 2 10 =1024 Log 2 1024=10

26. 11.4 Logarithmic Functions Recall exponent property A u = A v then U = V Evaluate log 7 1/49 – Let x = log 7 1/49 – Rewrite as exponential – 7 x =1/49 – 7 x =(49) -1 – 7 x =(7 2 ) -1 – 7 x =7 -2 – X=-2

27. 11.4 Logarithmic Functions Evaluate log 5 1/625 5 x =1/625 X = -4

28. 11.4 Logarithmic Functions Properties of Logarithms – Product – Quotient – Power – Equality

29. 11.4 Logarithmic Functions Solve each equation – Log p 64 1/3 =1/2 64 1/3 =p 1/2 3 √64= √ p 4= √ p P=16

30. 11.4 Logarithmic Functions Log 4 (2x+11)= Log 4 (5x-4) – 2x+11=5x-4 – X = 5 Log 11 x+log 11 (x+1)=Log 11 6 – Log 11 [x(x+1)] =Log 11 6 – x 2 +x=6 – X=2, -3

31. 11.5 Common Logarithms Common Logarithms – Logarithms with base 10 – When no base is given, 10 is assumed – Log 1000=3 ; 1000=10 3 – Log 1=0 ; 1=10 0 – Log 0.001=-3 ; 0.001=10 -3 – Log10 m =m

32. 11.5 Common Logarithms Given log 7 =0.8451 – Log 7,000,000 Log (1,000,000 * 7) Log 10 6 +log 7 6 + 0.8451 6.8451 – Log 0.0007 Log(0.0001*7) Log 10 -4 +log 7 -4+0.8451 -3.1549

33. 11.5 Common Logarithms Log5(2) 3 – Log5 + 3Log2 – 0.6990 + 3(0.3010) – 1.6021 Log13 3 /7 – 3log13-log7 – 2.4967

34. 11.5 Common Logarithms Change of Base for Logs – Log a n= log b n log b a Log 9 1043 – Log 10 1043 Log 10 9 3.1630

35. 11.5 Common Logarithms 6 3x =81 – Log6 3x =log81 – 3x*log6=log81 – 3x=log81 log 6 X=0.8175 5 4x =73 – 0.6665

36. 11.5 Common Logarithms 5 x-1 =2 x – X = 1.76

37. 11.6 Natural Logarithms Ln(x) – Log with base e – Log e x=lnx – Ln(e) = 1, very important property

38. 11.6 Natural Logarithms Convert log 6 254 to a natural logarithm and evaluate – Log 6 254= Log e 254 Log e 6 =ln254 ln6 3.0904

39. 11.6 Natural Logarithms Solve – 7.2 = -28.8 *ln (x) – -.25=ln(x) – Raise each side to the power of e – e -.25 = x – X = 0.78 Solve – 3.75=-7.5lnx – 0.6065=x

40. 11.6 Natural Logarithms 3 2x =7 x-1 – X=-7.7433 5 2x =7 x+1 – X=1.5287 4 x 2 -1 <12 – -1.67<X<1.67

41. 11.6 Natural Logarithms 3 2x =7 x-1 – X=-7.7433 5 2x =7 x+1 – X=1.5287 4 x 2 -1 <12 – -1.67<X<1.67

42. 11.7 Doubling Time The amount of time t required for a quantity to double is given by – T = Ln 2 k

43. 11.7 Doubling Time What interest rate is required for an investment with continuously compounded interest to double in 4 years? – T = Ln2 k T = Ln2 4 T = 0.1733