Logarithmic Functions & Their Graphs

1. Logarithmic Functions & Their Graphs Section 3.2

2. Log Functions & Their Graphs In the previous section, we worked with exponential functions. What did the graph of these functions look like?

3. Log Functions & Their Graphs Earlier in the year, we covered “inverse functions” Do exponential functions have an inverse? By looking at the graphs of exponential functions, we notice that every graph passes the horizontal line test. Therefore, all exponential functions have an inverse

4. Log Functions & Their Graphs The inverse of an exponential function with base a is called the logarithmic function with base a For x > 0, a > 0 and a ≠ 1

5. Log Functions & Their Graphs • In other words: really means that a raised to the power of y is equal to x • The log button on your calculator refers to the Log base 10 • This is referred to as the Common Logarithm

6. Log Functions & Their Graphs Another common logarithm is the Log base e This is referred to as the Natural Logarithmic Function This function is denoted:

7. Log Functions & Their Graphs Write the following logarithms in exponential form.

8. Log Functions & Their Graphs Write the exponential equations in log form

9. Log Functions & Their Graphs Evaluate the following logarithms: Since a raised to the power of zero is equal to 1, Since a raised to the power of one is equal to a = 0 = 1

10. Log Functions & Their Graphs Now that we know the definition of a logarithmic function, we can start to evaluate basic logarithms. What is this question asking? 2 raised to what power equals 8? 2³= 8 x = 3

11. Log Functions & Their Graphs Evaluate the following logarithms:

12. Log Functions & Their Graphs Properties of Logarithms

13. Log Functions & Their Graphs Using these properties, we can simplify different logarithmic functions. = x From our third property, we can evaluate this log function to be equal to x.

14. Log Functions & Their Graphs Use the properties of logarithms to evaluate or simplify the following expressions.

15. Log Functions & Their Graphs In conclusion, what does the following statement mean? “10 raised to the power of y is equal to z”

16. Logarithmic Functions & Their Graphs Section 3.2

17. Log Functions & Their Graphs Yesterday, we went over the basic definition of logarithms. Remember, they are truly defined as the inverse of an exponential function.

18. Graphs of Log Functions Fill in the following table and sketch the graph of the function f(x) for: f(x) =

19. Graphs of Log Functions Remember that the function is actually the inverse of the exponential function To graph inverses, switch the x and y values This is a reflection across the line y = x

20. Graphs of Log Functions Fill in the following table and sketch the graph of the function f(x) for:

21. Graphs of Log Functions The nature of this curve is typical of the curves of logarithmic functions. They have one x-intercept and one vertical asymptote Reflection of the exponential curve across the line y = x

22. Graphs of Log Functions Basic characteristics of the log curves Domain: (0, ∞) Range: (- ∞, ∞) x-intercept at (1, 0) Increasing 1-1 → the function has an inverse y-axis is a vertical asymptote Continuous

23. Graphs of Log Functions Much like we had shifts in exponential curves, the log curves have shifts and reflections as well Graphing will shift the curve 1 unit to the right Graphing will shift the curve vertically up 2 units

24. Graphs of Log Functions Much like we had shifts in exponential curves, the log curves have shifts and reflections as well Graphing will reflect the curve over the vertical asymptote Graphing will reflect the curve over the x-axis

25. Graphs of Log Functions Sketch a graph of the following functions.

26. Graphs of Log Functions Domain: (3, ∞) x-intercept: (4, 0) Asymptote: x = 3

27. Graphs of Log Functions Domain: (1, ∞) x-intercept: ( , 0) Asymptote: x = 1

28. Graphs of Log Functions Domain: (- ∞, 3) x-intercept: (2, 0) Asymptote: x = 3

29. Graphs of Log Functions Notice that the first piece of information we have been gathering on the graphs is the domain. For x > 0, a > 0 and a ≠ 1 This means that whatever value is in the place of x must be positive

30. Graphs of Log Functions What would the domain of this function be? (0, ∞) → - x > 0 → x < 0 → The domain would be: (-∞, 0)

31. Graphs of Log Functions Find the domain of the following logarithms. a) b) c)

32. Applications The model below approximates the length of a home mortgage of $150,000 at 8% interest in terms of the monthly payment. In the model, t is the number of years of the mortgage and x is the monthly payment in dollars.

33. Applications Use this model to approximate the length of a mortgage if the monthly payment is $1,300. By putting $1,300 in for x, you should get a time of 18.4 years

34. Applications • How much would you end up paying in interest using this same example? • Paying $1,300 a month for 18.4 years → Pay a total of (18.4) (1,300) = • Therefore, interest would be equal to $137,040. $287,040

35. Applications Using this same model, approximate the length of a mortgage when the monthly payment is: a) $1,100.65 and b) $1,254.68

36. Applications a) $1,100.65 and b) $1,254.68 What would the difference in amount paid be for each of these mortgages? 30 years 20 years

37. Applications A principal P, invested at 6% interest compounded continuously, increases to an amount K times the original principal after t years, where t is given by: How long will it take the original investment to double? By putting in 2 for K, we get t = 11.55 years