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Lesson 7.4 , For use with pages 499-505

Lesson 7.4 , For use with pages 499-505. WARMUP. ANSWER. y = x +. 1. 5. about $10,618.37. 4. ANSWER. ANSWER. 3. 3. Find the inverse of the function y = 3 x – 5 . Note: To get an inverse, r eplace the x with a y and vice versa. Then, get y by itself on one side. Or.

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Lesson 7.4 , For use with pages 499-505

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  1. Lesson 7.4, For use with pages 499-505 WARMUP ANSWER y = x + 1 5 about $10,618.37 4 ANSWER ANSWER 3 3 • Find the inverse of the function y = 3x – 5. Note: To get an inverse, replace the x with a y and vice versa.Then, get y by itself on one side. Or 2.An account that pays 3% annual interest compounded continuously has a balance of $10,000 on June 1, 2008. If no money is added, what is the balance on June 1, 2010? Note: Use the formula A = Pert A = Amount you end up with. P = Principal (starting amount) e  2.72, r = rate as a decimal, t = time in years 3. 2 to what power equals 16? 4. How much more powerful is an 8 earthquake compared to a 4?

  2. 7.1 – 7.3 • Look at Exponential Growth and Decay graphs

  3. 7.4 Notes - Evaluate Logarithms and Graph Log Functions Aka: Logs http://www.youtube.com/watch?v=eusMzC7Rx7M&feature=related

  4. Dictionary.com • log • 1    lɔg, lɒg/ Show Spelled [lawg, log] Show IPA noun, verb, logged, log·ging. • noun • 1. • a portion or length of the trunk or of a large limb of a felled tree. • 2. • something inert, heavy, or not sentient. • 3. • Nautical . any of various devices for determining the speed of a ship, as a chip log  or patent log. • 4. • any of various records, made in rough or finished form, concerning a trip made by a ship or aircraft and dealing with particulars of navigation, weather, engine performance, discipline, and other pertinent details; logbook. • 5. • Movies . an account describing or denoting each shot as it is taken, written down during production and referred to in editing the film.

  5. Logger Days

  6. Actual Logger

  7. Objective - To evaluate and graph logarithmic functions. Rewriting Logarithmic Equations.

  8. Evaluate. = x 1) 3) = x 2) 4) = x = x

  9. Find the inverse of Switch x and y. Write in exponential form. Find the inverse of Switch x and y. Write in exponential form. Solve for y.

  10. Graph y x Domain: x > 0 Range: all reals

  11. Graph y x Domain: x > 2 Range: all reals

  12. log log log log a. 8 3 23 8 = = 4 12 2 1/4 b. 1 0 40 1 = = c. 12 1 121 12 = = –1 1 d. 4 –1 4 = = 4 EXAMPLE 1 Rewrite logarithmic equations Logarithmic Form Exponential Form

  13. log log log log 3 7 1/2 14 1. 81 4 34 81 = = 2. 7 1 71 7 = = –5 1 3. 1 0 140 1 = = 2 4. 32 –5 32 = = for Example 1 GUIDED PRACTICE Rewrite the equation in exponential form. Logarithmic Form Exponential Form

  14. log log log log b. 0.2 a. 64 b 5 4 4 To help you find the value of y, ask yourself what power of bgives you y. a. 4 to what power gives 64? 43 64, so 64 3. = = b. 5 to what power gives 0.2? 0.2, so 0.2 –1. 5–1 log = = 5 EXAMPLE 2 Evaluate logarithms Evaluate the logarithm. SOLUTION

  15. log log log log log d. 125 c. 6 1/5 36 1/5 b 36 –3 1 1 To help you find the value of y, ask yourself what power of bgives you y. 5 5 c. to what power gives 125? 125, so 125 –3. = = 1 361/2 6, so 6 . d. 36 to what power gives 6? = = 2 EXAMPLE 2 Evaluate logarithms Evaluate the logarithm. SOLUTION

  16. log 8 a. 8 100.903 8 b. ln 0.3 .3 e –1.204 0.3 EXAMPLE 3 Evaluate common and natural logarithms Expression Keystrokes Display Check 0.903089987 –1.203972804 Do this one!!!!

  17. Tornadoes 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. EXAMPLE 4 Evaluate a logarithmic model

  18. s 93logd+ 65 = 93(2.342)+ 65 ANSWER The wind speed near the tornado’s center was about 283miles per hour. EXAMPLE 4 Evaluate a logarithmic model SOLUTION Write function. = 93log220+ 65 Substitute 220 for d. Use a calculator. = 282.806 Simplify.

  19. log log 5. 32 27 2 log 12 7. 8. ln 0.75 6. 3 1 3 for Examples 2, 3 and 4 GUIDED PRACTICE Evaluate the logarithm. Use a calculator if necessary. SOLUTION 5 SOLUTION 1.079 SOLUTION SOLUTION –0.288

  20. 9. WHAT IF?Use the function in Example 4 to estimate the wind speed near a tornado’s center if its path is 150miles long. ANSWER The wind speed near the tornado’s center is about 267 miles per hour. for Examples 2, 3 and 4 GUIDED PRACTICE

  21. 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 EXAMPLE 5 Use inverse properties Simplify the expression. SOLUTION Express 25 as a power with base 5. Power of a power property

  22. 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. EXAMPLE 6 Find inverse functions 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.

  23. log 10. 8 log x 8 7 x 7–3x 11. –3x for Examples 5 and 6 GUIDED PRACTICE Simplify the expression. SOLUTION SOLUTION

  24. log 12. 64x 2 6x 13. eln20 for Examples 5 and 6 GUIDED PRACTICE Simplify the expression. SOLUTION SOLUTION 20

  25. 14. Find the inverse of y = 4 x log 4 y = x. 15. Find the inverse of y =ln(x – 5). y = ex + 5. for Examples 5 and 6 GUIDED PRACTICE SOLUTION SOLUTION

  26. log a. y x = 3 EXAMPLE 7 Graph logarithmic functions 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.

  27. log b. y x = 1/2 EXAMPLE 7 Graph logarithmic functions Graph the function. SOLUTION Plot several convenient points, such as (1, 0), (2, –1), (4, –2), and (8, –3). 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 down through the plotted points, as shown below.

  28. Graph . State the domain and range. y (x + 3) + 1 = log log 2 2 Sketch the graph of the parent function y = x, which passes through (1, 0), (2, 1), and (4, 2). EXAMPLE 8 Translate a logarithmic graph SOLUTION STEP 1 STEP 2 Translate the parent graph left 3 units and up 1 unit. The translated graph passes through (–2, 1), (–1, 2), and (1, 3). The graph’s asymptote is x = –3. The domain is x > –3, and the range is all real numbers.

  29. log 16. y x = 5 for Examples 7 and 8 GUIDED PRACTICE Graph the function. State the domain and range. SOLUTION The domain is x > 0, and the range is all real numbers.

  30. log 17. y (x – 3) = 1/3 for Examples 7 and 8 GUIDED PRACTICE Graph the function. State the domain and range. SOLUTION domain: x > 3, range: all real numbers

  31. log 18. y (x + 1) – 2 = 4 for Examples 7 and 8 GUIDED PRACTICE Graph the function. State the domain and range. SOLUTION domain: x > –1, range: all real numbers

  32. 7.4 Assignment 7.4: 1-35 ODD, 65-79 ODD

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