4.4 Evaluate Logarithms & Graph Logarithmic Functions

1. 4.4Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents? What is the definition? How do you rewrite log equations? What are two special log values? What is a common log? A natural log? What logs can you evaluate using a calculator?

2. Evaluating Log Expressions • We know 22 = 4 and 23 = 8 • But for what value of y does 2y = 6? • Because 22<6<23 you would expect the answer to be between 2 & 3. • To answer this question exactly, mathematicians defined logarithms.

3. Definition of Logarithm to base b • Let b & x be positive numbers & b ≠ 1. • The logarithm of x with base a is denoted by logbx and is defined: • logbx = y if by = x • This expression is read “log base b of x” • The function f(x) = logbx is the logarithmic function with base b.

4. The definition tells you that the equations logbx = y and by = x are equivalent. • Rewriting forms: • To evaluate log3 9 = x ask yourself… • “Self… 3 to what power is 9?” • 32 = 9 so…… log39 = 2

5. log216 = 4 log1010 = 1 log31 = 0 log10 .1 = -1 log2 6 ≈ 2.585 24 = 16 101 = 10 30 = 1 10-1 = .1 22.585 = 6 Log formExp. form

6. log381 = Log5125 = Log4256 = Log2(1/32) = 3x = 81 5x = 125 4x = 256 2x = (1/32) Evaluate 4 3 4 -5

7. Evaluating logarithms now you try some! 2 • Log 4 16 = • Log 5 1 = • Log 4 2 = • Log 3 (-1) = • (Think of the graph of y=3x) 0 ½ (because 41/2 = 2) undefined

8. You should learn the following general forms!!! • Log b 1 = 0 because b0 = 1 • Log b b = 1 because b1 = b • Log b bx = x because bx = bx

9. Natural logarithms • log e x = ln x • ln means log base e e

10. Common logarithms • log 10 x = log x • Understood base 10 if nothing is there.

11. Common logs and natural logs with a calculator log10 button ln button **Only common log and natural log bases are on a calculator.

12. log 8 a. 8 100.903 8 b. ln 0.3 .3 e –1.204 0.3 Keystrokes Expression Keystrokes Display Check 0.903089987 –1.203972804

13. The wind speed s(in miles per hour) near the center of a tornado can be modeled by: s 93logd + 65 = where dis the distance (in miles) that the tornado travels. In 1925, a tornado traveled 220 miles through three states. Estimate the wind speed near the tornado’s center. Tornadoes

14. s 93logd+ 65 = 93(2.342)+ 65 ANSWER The wind speed near the tornado’s center was about 283 miles per hour. Solution Write function. = 93log220+ 65 Substitute 220 for d. Use a calculator. = 282.806 Simplify.

15. What is a logarithm? How do you read it? A logarithm is another way of expressing an exponent. It is read log base b of y. • What relationship exists between logs and exponents? What is the definition? • logax = y if ay = x • How do you rewrite logs? The base with the exponent on the other side of the = . • What are two special log values? Logb1=0 and logbb=1 • What is a common log? A natural log? Common log is base 10. Natural log is base e. • What logs can you evaluate using a calculator? Base 10

16. 4.4 Assignment Page 255, 3-6, 8-16, 20-26 even

17. 4.4 Day 2 • How do you use inverse properties with logarithms? • How do you graph logs?

18. g(x) = log b x is the inverse of • f(x) = bx • f(g(x)) = x and g(f(x)) = x • Exponential and log functions are inverses and “undo” each other.

19. So: g(f(x)) = logbbx = x • f(g(x)) = blogbx = x • 10log2 = • Log39x = • 10logx = • Log5125x = 2 Log3(32)x = Log332x= 2x x 3x

20. log log log a. b. 25x 10log4 5 5 5 4 a. 10log4 = b log x = x b ( ) log 25x b. 52 x = 5 52x = 2x = bx log = x b Use Inverse Properties Simplify the expression. SOLUTION Express 25 as a power with base 5. Power of a power property

21. y = ln (x + 3) log a. b. y = 6 x 6 a. From the definition of logarithm, the inverse of is x. y = 6 y= x ex = (y + 3) ex – 3 y = ANSWER The inverse ofy =ln(x + 3) isy = ex – 3. Find Inverse Properties Find the inverse of the function. SOLUTION b. y = ln (x + 3) Write original function. x = ln (y + 3) Switch x and y. Write in exponential form. Solve for y.

22. log log 10. 8 log x 8 7 7 8 x = b = x log x log b 8 b 7–3x 11. = x log ax –3x 7–3x = a Use Inverse Properties Simplify the expression. SOLUTION Exponent form Log form SOLUTION Log form Exponent form

23. 14. Find the inverse of y = 4 x log 4 From the definition of logarithm, the inverse of y = x. is 15. Find the inverse of y =ln(x – 5). ex = (y – 5) ex + 5 y = The inverse ofy =ln(x – 5) isy = e x + 5. ANSWER Use Inverse Properties SOLUTION SOLUTION y =ln(x – 5) Write original function. x = ln (y – 5) Switch x and y. Write in exponential form. Solve for y.

24. Finding Inverses • Find the inverse of: • y = log3x • By definition of logarithm, the inverse is y=3x • OR write it in exponential form and switch the x & y! 3y = x 3x = y

25. Finding Inverses cont. • Find the inverse of : • Y = ln (x +1) • X = ln (y + 1) Switch the x & y • ex = y + 1 Write in exp form • ex – 1 = y solve for y

26. 4.4 Graphing Logs p. 254

27. Graphs of logs • y = logb(x-h)+k • Has vertical asymptote x=h • The domain is x>h, the range is all reals • If b>1, the graph moves up to the right • If 0<b<1, the graph moves down to the right

28. log a. y x = 3 Graphing a log function Graph the function. SOLUTION Plot several convenient points, such as (1, 0), (3, 1), and (9, 2). The y-axis is a vertical asymptote. From left to right, draw a curve that starts just to the right of the y-axis and moves up through the plotted points, as shown below.

29. Graph y = log1/3x-1 • Plot (1/3,0) & (3,-2) • Vert line x=0 is asy. • Connect the dots X=0

30. Plot easy points (-1,0) & (3,1) Label the asymptote x=−2 Connect the dots using the asymptote. Graph y =log5(x+2) X=-2

31. How do you use inverse properties with logarithms? Exponential and log functions are inverses and “undo” each other. • How do you graph logs? Pick 1, the base number, and a power of the base for x.

32. 4.4 Assignment Day 2 • Page 256, 28-44 even, 45-51 odd