1. Logarithmic Functions Objectives: Change Exponential Expressions <-  Logarithmic Expressions Evaluate Logarithmic Expressions Determine the domain of a logarithmic function Graph and solve logarithmic equations

2. Logarithmic Functions Inverse of Exponential functions: If a x = y, then log a y = x Domain: 0 < x < infinity Range: neg. infinity < y < infinity

3. Translate each of the following to logarithmic form. 2 3 = 8 4 1/2 = 2 Find the domain of: F(x) = log 2 (x – 5) G(x) = log 5 ((1+x)/(1-x))

4. To graph logarithmic functions Graph the related exponential function. Reflect this graph across the y=x line (Switch the x’s and y’s) Graph: y = log 1/3 x

5. Natural logarithms and Common Logarithms Natural Logarithm (ln) : log e Common Logarithm (log): log 10 Graph y=ln x (Reflect the graph of y=e x ) Graph y = -ln (x + 2), Determine the domain, range, and vertical asymptote. Describe the translations.

6. Graph: f(x) = log x (Reflect the graph of y = 10 x ) Graph: f(x) = 3 log (x – 1). Determine the domain, range, and vertical asymptote. Describe the translations on the graph

7. Solving Logarithmic Equations Logarithm on one side: Write equation in exponential form and solve Examples: Solve: log 3 (4x – 7) = 2 Solve: log 2 (2x + 1) = 3

8. Example The atmospheric pressure ‘p’ on a balloon or an aircraft decreases with increasing height. This pressure, measured in millimeters of mercury, is related to the height ‘h’ (in kilometers) above sea level by the formula p=760e -0.145h Find the height of an aircraft if the atmospheric pressure is 320 millimeters of mercury.

9. Example 2 The loudness L(x), measure in decibels, of a sound of intensity x, measure in watts per square meter, is defined as L(x)=10log(x/I o ) where I o = 10 -12 watt per square meter is the least intense sound that a human ear can detect. Determine the loudness, in decibels, of heavy city traffic: intensity of x=10 -3 watt per square meter.

10. Example 3 Richter Scale: M(x) = log (x/x o ) where x 0 =10 -3 is the reading of a zero-level earthquake the same distance from its epicenter. Determine the magnitude of the Mexico City earthquake in 1985: seismographic reading of 125,892 millimeters 100 kilometers from the center.

11. Properties of Logarithms Log a 1 = 0 Log a a = 1 a logaM = M Log a a r = r

12. Log a (MN) = log a M + log a N Log a (M/N) = log a M – log a N Log a M r = r log a M

13. Look at Examples Page 444-445 Other examples: Page 449: #8, 12, 16, 20, 24, 28, 32, 36, 44, 52, 60

14. Change of Base Formula: log a M= log b M / log b a Example: log 5 89 Example: log 6 32 Page 449: #65, 71, 74

15. Solving logarithmic equations With logarithms on both sides. Combine each side to one logarithm Cancel the logarithms out Solve the remaining equation Examples: Page 450: #81, 87

16. Logarithm on One side of Equation Combine terms into one logarithm Write in exponential form Solve equation that will form Ex: Page 454 #33, 37

17. Solving Exponential Equations Variable is in the exponent. Use logarithms to bring exponent down and solve. Solve: 4 x – 2 x – 12 = 0 Solve: 2 x = 5

18. Solve: 5 x-2 = 3 3x+2 log 3 x + log 3 8 = -2 8. 3 x = 5 log 3 x + log 4 x = 4

19. Applications Simple Interest: I = Prt Interest = Principal X Rate X time Compount Interest: A = P. (1 + r/n) nt Time is in years Annually: once a year Semiannually: Twice per year Quarterly: Four times per year Monthly: 12 times per year Daily: 365 times per year

20. Compound Continuously Interest A = Pe rt The present value P of A dollars to be received after ‘t’ years, assuming a per annum interest rate ‘r’ compounded ‘n’ times per year, is P=A. (1 + r/n) -nt

21. Finding Effective Rate of Interest On January 2, 2004, $2000 is placed in an Individual Retirement Account (IRA) that will pay interest of 10% per annum compounded continuously. What will the IRA be worth on January 1, 2024? What is the effective rate of interest?

22. Present Value Formula for compounded continuously interest P = A( 1 + r/n) -nt P = Ae -rt Examples: Page 462 #5, 11, 15, 21

23. Exponential Decay P = Ae -rt Page 472 #3

24. Other Applications A(t) = A o e kt : Exponential Growth Newton’s Law of Cooling: U(t) = T + (u o – T)e kt, k < 0 Logistic Growth Model: P(t) = c / (1 + ae -bt ) c: carrying capacity

25. Examples Page 472: #1, 13, 22

26. Assignment Page 454, 462, 472

27. Exponential and Logarithmic Regressions Input data into calculator Go to calculate mode Find ExpReg (Exponential Regression) y = ab x Find LnReg (natural logarithm regression) y = a + b. lnx Logistic Regression y=c/(1+ae -bx )

28. Examples Page 479: #1, 3, 7, 11 1. b. EXP REG: y =.0903(1.3384) x c. y=..0903(e ln(1.3384) ) x d. Graph: y =.0903e.2915x e. n(7) =.0903e (.2915 x 7) f..0903e (.2915(t)) =.75

29. 3. b. EXP REG: y = 100.3263(.8769) x c. 100.3263(e ln.8769 ) x d. Graph: y = 100.3263e (-.1314)x e. 100.3263e (-.1314)x =.5 (100.3263) f. 100.3263e (.1314)(50) =.141 g. 100.3263e (-.1314)x = 20

30. 7. b. LnReg: y = 32741.02 – 6070.96lnx c. Graph d. 1650 = 32741.02 – 6070.96 lnx = 168 computers

31. 11. b. LOGISTIC REG (not all calculators have): Y = 14471245.24 / (1 + 2.01527e -.2458x ) c. Graph d. Y = 14,471,245.24 / (1 + 2.01527e -.2458x ) Y = 14,471,245.24 / (1 + 0) e. 12.750,854 = 14,471,245.24 / (1 + 2.01527e -.2458x )

32. Assignment Pages: 472, 479