8.4 Logarithmic Functions

1. 8.4 Logarithmic Functions Objectives: 1. Write logarithmic function in exponential form and back 2. Evaluate logs with and without calculator 3. Evaluate the logarithmic function 4. Understand logs and inverses 5. Graph logarithmic function Vocabulary: logarithm, common logarithm, natural logarithm

2. In last section, 8.3, we have learned that the Interest problem that if the interest of a bank account of initial asset P is 5% compounded and the total asset after t years is described by the model (exponential growth): Yearly: At= P (1 + 0.05 )t Monthly: At= P (1 + 0.05 / 12)12·t Daily: At= P (1 + 0.05 / 365)365·t Continuously: At= Pe 0.05·t

3. In each case, as long as we know the duration of the asset deposited in the bank, the t, we can calculate the final (total) asset: Yearly: A5= 1200 (1 + 0.05 )5 Monthly: A10= 3500 (1 + 0.05 / 12)12·10 Daily: A2= 9560 (1 + 0.05 / 365)365·2 Continuously: A6= 27890e 0.05·6 Now we would like to ask a reverse question: How long does the initial deposit (investment) take to reach the target asset value?

4. 4 to what power gives me 64? 3 to what power gives me 81?

5. 2 to what power gives me .125? 1/4 to what power gives 256?

6. 32 to what power gives me 2?

7. DefinitionExponential Function The function is of the form: f (x) = a bx, where a≠ 0, b > 0 and b ≠ 1, x  R. Simple Exponential Function Let b and y be positive numbers, and b ≠ 1. y = b x DefinitionLogarithm of y with base b Let b and y be positive numbers, and b ≠ 1. logby = x if and only if y = b x

8. Example 1 • log39 • log41 c) log5(1/25)

9. Special Logarithm Values • logb1 = 0 • logbb = 1 Note: Some other special logarithm values are: • logb0 = undefined d) logb (-2) = undefined

10. Challenge Question One student said since (-3)2 = 9, so, log(-3)9 = 2, why do we need to constrain the base to be positive in the definition? Answer to this Question If the negative number can be used as a base, when we are going to discuss the more general situation such as log-33, this will turn out 3 = (-3)x. And this will never be true for any real number x.

11. Practice • log13169 • log1001 • log25252 d) log25255

12. Example 2 Evaluate the expression • log464 • log20.125 c) log1/4256 • log322

13. Practice Evaluate the expression • log327 • log40.0625 c) log1/16256 • log642 e) If k > 0 and k ≠ 1, logk1

14. Definition Common Logarithm log10x = logx Definition Natural Logarithm logex = lnx

15. Example 3 Evaluate the common and natural logarithm • log4 • ln(1/5) c) lne-3 • log(1/1000)

16. Practice Evaluate the common and natural logarithm • log7 • ln0.25 c) log3.8 • ln3 e) lne2007

17. The logarithmic function with base b is defined as g(x) = logbx with domain x  R+.

18. From the definition of a logarithm, we noticed that the logarithmic function g(x) = logbx is the inverse of the exponential function f(x) = bx Because This means that they offset each other, or they are “undo” each other.

19. Example 4 The Richter scale is used for measuring the magnitude of an earthquake. The Richter magnitude R is given by the model R = 0.67 log(0.37E) + 1.46 Where E is the energy (in kilowatt-hours) released by the earthquake. • Suppose an earthquake releases 15,500, 000,000 million kwh of energy. What is the earthquake’s magnitude? (7.998) b) How many kwh of energy would the earthquake above have to release in order to increase its magnitude by one-half of a unit on the Richter scale? (8.6417E10)

20. Example 5 Evaluate the common and natural logarithm • 10log4 • eln(1/5) c) log5125 x • lne – 3 x

21. Practice Simply • 10log5x • log100002x

22. From the definition of a logarithm, we noticed that the logarithmic function g(x) = logbx is the inverse of the exponential function f(x) = bx Because This means that they offset each other, or they are “undo” each other. These two functions are inverse to each other.

23. The graph of the logarithmic function f(x) = bx ( b > 1) is x-axis and y-axis are horizontal and vertical asymptotes. the graph of its inverse function g(x) = logbx Two graphs are symmetry to the line y = x

24. The graph of the logarithmic function f(x) = bx ( 0 < b < 1) is x-axis and y-axis are horizontal and vertical asymptotes. the graph of its inverse function g(x) = logbx Two graphs are symmetry to the line y = x

25. Example 6 Find the inverse of the function • y = log8x • y = ln(x – 3) Answer • y = 8x • y = ex + 3

26. Practice Find the inverse of • y = log2/5x • y = ln(2x – 10) Answer • y = (2/5)x • y = (ex + 10)/2

27. Function Family The graph of the function y = f(x – h)  k x – h = 0, x = h is the graph of the function y = f(x) shift h unit to the right and k unit up/down. The graph of the function y = f(x + h) k x + h = 0, x = –h is the graph of the function y = f(x) shift h unit to the left and k unit up/down.

28. Logarithmic Function Family The graph of the logarithmic function y = logb(x h)  k has • The domain is x > –h (x > h). • The line x = –h (x = h) is the vertical asymptote. • If 0 < b < 1, the curve goes down. If b > 1, the curve goes up. 4) The function graph is shifted h units horizontally and k units vertically from the graph y = logbx

29. Example 7 Graph the function, state domain and range. • y = log1/2 (x + 4) + 2 b) y = log3(x – 2) – 1 0 1 2 - 4 0 1

30. 8.4 Logarithmic Functions Assignment: 8.4 P490 #17-76 even - Show work