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# 3.8 calculation with log and exp

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### 3.8 calculation with log and exp

1. 1. Calculation with Log and Exp http://www.lahc.edu/math/precalculus/math_260a.html
2. 2. In this section, we solve simple numerical equations involving log and exponential functions in base 10 or base e. Calculation with Log and Exp
3. 3. In this section, we solve simple numerical equations involving log and exponential functions in base 10 or base e. Most numerical calculations in science are in these two bases. Calculation with Log and Exp
4. 4. In this section, we solve simple numerical equations involving log and exponential functions in base 10 or base e. Most numerical calculations in science are in these two bases. We need a calculator that has the following functions: ex, 10x, ln(x), and log(x). Calculation with Log and Exp
5. 5. In this section, we solve simple numerical equations involving log and exponential functions in base 10 or base e. Most numerical calculations in science are in these two bases. We need a calculator that has the following functions: ex, 10x, ln(x), and log(x). All answers are given to 3 significant digits. Calculation with Log and Exp
6. 6. In this section, we solve simple numerical equations involving log and exponential functions in base 10 or base e. Most numerical calculations in science are in these two bases. We need a calculator that has the following functions: ex, 10x, ln(x), and log(x). All answers are given to 3 significant digits. 6 Example A: Find the answers with a calculator. a.103.32 b. e = e1/6 c. log(4.35) d. ln(2/3) Calculation with Log and Exp
7. 7. In this section, we solve simple numerical equations involving log and exponential functions in base 10 or base e. Most numerical calculations in science are in these two bases. We need a calculator that has the following functions: ex, 10x, ln(x), and log(x). All answers are given to 3 significant digits. 6 Example A: Find the answers with a calculator. a.103.32 b. e = e1/6  2090 c. log(4.35) d. ln(2/3) Calculation with Log and Exp
8. 8. In this section, we solve simple numerical equations involving log and exponential functions in base 10 or base e. Most numerical calculations in science are in these two bases. We need a calculator that has the following functions: ex, 10x, ln(x), and log(x). All answers are given to 3 significant digits. 6 Example A: Find the answers with a calculator. a.103.32 b. e = e1/6  2090  1.18 c. log(4.35) d. ln(2/3) Calculation with Log and Exp
9. 9. In this section, we solve simple numerical equations involving log and exponential functions in base 10 or base e. Most numerical calculations in science are in these two bases. We need a calculator that has the following functions: ex, 10x, ln(x), and log(x). All answers are given to 3 significant digits. 6 Example A: Find the answers with a calculator. a.103.32 b. e = e1/6  2090  1.18 c. log(4.35) d. ln(2/3) 0.638 Calculation with Log and Exp
10. 10. In this section, we solve simple numerical equations involving log and exponential functions in base 10 or base e. Most numerical calculations in science are in these two bases. We need a calculator that has the following functions: ex, 10x, ln(x), and log(x). All answers are given to 3 significant digits. 6 Example A: Find the answers with a calculator. a.103.32 b. e = e1/6  2090  1.18 c. log(4.35) d. ln(2/3) 0.638  -0.405 Calculation with Log and Exp
11. 11. In this section, we solve simple numerical equations involving log and exponential functions in base 10 or base e. Most numerical calculations in science are in these two bases. We need a calculator that has the following functions: ex, 10x, ln(x), and log(x). All answers are given to 3 significant digits. 6 Example A: Find the answers with a calculator. a.103.32 b. e = e1/6  2090  1.18 c. log(4.35) d. ln(2/3) 0.638  -0.405 These problems may be stated in alternate forms. Calculation with Log and Exp
12. 12. Example B: Find the x a. log(x) = 3.32 b. 1/6 = ln(x) c. 10x = 4.35 d. 2/3 = ex Calculation with Log and Exp
13. 13. Example B: Find the x a. log(x) = 3.32 b. 1/6 = ln(x) x =103.32 ( 2090) c. 10x = 4.35 d. 2/3 = ex Calculation with Log and Exp
14. 14. Example B: Find the x a. log(x) = 3.32 b. 1/6 = ln(x) x =103.32 ( 2090) e1/6 = x ( 1.18) c. 10x = 4.35 d. 2/3 = ex Calculation with Log and Exp
15. 15. Example B: Find the x a. log(x) = 3.32 b. 1/6 = ln(x) x =103.32 ( 2090) e1/6 = x ( 1.18) c. 10x = 4.35 d. 2/3 = ex x = log(4.35) ( 0.638) Calculation with Log and Exp
16. 16. Example B: Find the x a. log(x) = 3.32 b. 1/6 = ln(x) x =103.32 ( 2090) e1/6 = x ( 1.18) c. 10x = 4.35 d. 2/3 = ex x = log(4.35) ( 0.638) ln(2/3) = x ( -0.405) Calculation with Log and Exp
17. 17. Example B: Find the x a. log(x) = 3.32 b. 1/6 = ln(x) x =103.32 ( 2090) e1/6 = x ( 1.18) c. 10x = 4.35 d. 2/3 = ex x = log(4.35) ( 0.638) ln(2/3) = x ( -0.405) An equation is called a log-equations if the unknown is in the log-function as in parts a and b above. Calculation with Log and Exp
18. 18. Example B: Find the x a. log(x) = 3.32 b. 1/6 = ln(x) x =103.32 ( 2090) e1/6 = x ( 1.18) c. 10x = 4.35 d. 2/3 = ex x = log(4.35) ( 0.638) ln(2/3) = x ( -0.405) An equation is called an exponential equations if the unknown is in the exponent as in parts c and d. An equation is called a log-equations if the unknown is in the log-function as in parts a and b above. Calculation with Log and Exp
19. 19. Example B: Find the x a. log(x) = 3.32 b. 1/6 = ln(x) x =103.32 ( 2090) e1/6 = x ( 1.18) c. 10x = 4.35 d. 2/3 = ex x = log(4.35) ( 0.638) ln(2/3) = x ( -0.405) An equation is called an exponential equations if the unknown is in the exponent as in parts c and d. An equation is called a log-equations if the unknown is in the log-function as in parts a and b above. To solve log-equations, drop the log and write the problems in exp-form. Calculation with Log and Exp
20. 20. Example B: Find the x a. log(x) = 3.32 b. 1/6 = ln(x) x =103.32 ( 2090) e1/6 = x ( 1.18) c. 10x = 4.35 d. 2/3 = ex x = log(4.35) ( 0.638) ln(2/3) = x ( -0.405) An equation is called an exponential equations if the unknown is in the exponent as in parts c and d. An equation is called a log-equations if the unknown is in the log-function as in parts a and b above. To solve log-equations, drop the log and write the problems in exp-form. To solve exponential equations, lower the exponents and write the problems in log-form. Calculation with Log and Exp
21. 21. More precisely, to solve exponential equations, Calculation with Log and Exp
22. 22. More precisely, to solve exponential equations, we I. isolate the exponential part that contains the x, Calculation with Log and Exp
23. 23. More precisely, to solve exponential equations, we I. isolate the exponential part that contains the x, II. bring down the exponents by writing it in log-form. Calculation with Log and Exp
24. 24. More precisely, to solve exponential equations, we I. isolate the exponential part that contains the x, II. bring down the exponents by writing it in log-form. Example C: Solve 25 = 7*102x Calculation with Log and Exp
25. 25. More precisely, to solve exponential equations, we I. isolate the exponential part that contains the x, II. bring down the exponents by writing it in log-form. Example C: Solve 25 = 7*102x Isolate the exponential part containing the x, 25/7 = 102x Calculation with Log and Exp
26. 26. More precisely, to solve exponential equations, we I. isolate the exponential part that contains the x, II. bring down the exponents by writing it in log-form. Example C: Solve 25 = 7*102x Isolate the exponential part containing the x, 25/7 = 102x Bring down the x by restating it in log-form: log(25/7) = 2x Calculation with Log and Exp
27. 27. More precisely, to solve exponential equations, we I. isolate the exponential part that contains the x, II. bring down the exponents by writing it in log-form. Example C: Solve 25 = 7*102x Isolate the exponential part containing the x, 25/7 = 102x Bring down the x by restating it in log-form: log(25/7) = 2x log(25/7) 2 = x Exact answer Calculation with Log and Exp
28. 28. More precisely, to solve exponential equations, we I. isolate the exponential part that contains the x, II. bring down the exponents by writing it in log-form. Example C: Solve 25 = 7*102x Isolate the exponential part containing the x, 25/7 = 102x Bring down the x by restating it in log-form: log(25/7) = 2x log(25/7) 2 = x  0.276 Exact answer Approx. answer Calculation with Log and Exp
29. 29. Example D: Solve 2.3*e2-3x + 4.1 = 12.5 Calculation with Log and Exp
30. 30. Example D: Solve 2.3*e2-3x + 4.1 = 12.5 Isolate the exp-part: 2.3*e2-3x + 4.1 = 12.5 Calculation with Log and Exp
31. 31. Example D: Solve 2.3*e2-3x + 4.1 = 12.5 Isolate the exp-part: 2.3*e2-3x + 4.1 = 12.5 2.3*e2-3x = 12.5 – 4.1 2.3*e2-3x = 8.4 Calculation with Log and Exp
32. 32. Example D: Solve 2.3*e2-3x + 4.1 = 12.5 Isolate the exp-part: 2.3*e2-3x + 4.1 = 12.5 2.3*e2-3x = 12.5 – 4.1 2.3*e2-3x = 8.4 e2-3x = 8.4/2.3 Calculation with Log and Exp
33. 33. Example D: Solve 2.3*e2-3x + 4.1 = 12.5 Isolate the exp-part: 2.3*e2-3x + 4.1 = 12.5 2.3*e2-3x = 12.5 – 4.1 2.3*e2-3x = 8.4 e2-3x = 8.4/2.3 Restate in log-form: 2 – 3x = ln(8.4/2.3) Calculation with Log and Exp
34. 34. Example D: Solve 2.3*e2-3x + 4.1 = 12.5 Isolate the exp-part: 2.3*e2-3x + 4.1 = 12.5 2.3*e2-3x = 12.5 – 4.1 2.3*e2-3x = 8.4 e2-3x = 8.4/2.3 Restate in log-form: 2 – 3x = ln(8.4/2.3) Solve for x: 2 – ln(8.4/2.3) = 3x Calculation with Log and Exp
35. 35. Example D: Solve 2.3*e2-3x + 4.1 = 12.5 Isolate the exp-part: 2.3*e2-3x + 4.1 = 12.5 2.3*e2-3x = 12.5 – 4.1 2.3*e2-3x = 8.4 e2-3x = 8.4/2.3 Restate in log-form: 2 – 3x = ln(8.4/2.3) Solve for x: 2 – ln(8.4/2.3) = 3x 2-ln(8.4/2.3) 3 = x Calculation with Log and Exp
36. 36. Example D: Solve 2.3*e2-3x + 4.1 = 12.5 Isolate the exp-part: 2.3*e2-3x + 4.1 = 12.5 2.3*e2-3x = 12.5 – 4.1 2.3*e2-3x = 8.4 e2-3x = 8.4/2.3 Restate in log-form: 2 – 3x = ln(8.4/2.3) Solve for x: 2 – ln(8.4/2.3) = 3x 2-ln(8.4/2.3) 3 = x  0.235 Calculation with Log and Exp
37. 37. Example D: Solve 2.3*e2-3x + 4.1 = 12.5 Isolate the exp-part: 2.3*e2-3x + 4.1 = 12.5 2.3*e2-3x = 12.5 – 4.1 2.3*e2-3x = 8.4 e2-3x = 8.4/2.3 Restate in log-form: 2 – 3x = ln(8.4/2.3) Solve for x: 2 – ln(8.4/2.3) = 3x 2-ln(8.4/2.3) 3 = x  0.235 Calculation with Log and Exp We solve log-equation in analogous fashion:
38. 38. Example D: Solve 2.3*e2-3x + 4.1 = 12.5 Isolate the exp-part: 2.3*e2-3x + 4.1 = 12.5 2.3*e2-3x = 12.5 – 4.1 2.3*e2-3x = 8.4 e2-3x = 8.4/2.3 Restate in log-form: 2 – 3x = ln(8.4/2.3) Solve for x: 2 – ln(8.4/2.3) = 3x 2-ln(8.4/2.3) 3 = x  0.235 Calculation with Log and Exp We solve log-equation in analogous fashion: I. isolate the log part that contains the x, II. drop the log by writing it in exp-form.
39. 39. Calculation with Log and Exp Example E: Solve 9*log(2x+1)= 7
40. 40. Calculation with Log and Exp Example E: Solve 9*log(2x+1)= 7 Isolate the log-part, log(2x+1) = 7/9
41. 41. Calculation with Log and Exp Example E: Solve 9*log(2x+1)= 7 Isolate the log-part, log(2x+1) = 7/9 Write it in exp-form 2x + 1 = 107/9
42. 42. Calculation with Log and Exp Example E: Solve 9*log(2x+1)= 7 Isolate the log-part, log(2x+1) = 7/9 Write it in exp-form 2x + 1 = 107/9 Save for x:
43. 43. Calculation with Log and Exp Example E: Solve 9*log(2x+1)= 7 Isolate the log-part, log(2x+1) = 7/9 Write it in exp-form 2x + 1 = 107/9 Save for x: 2x = 107/9 – 1 x = (107/9 – 1)/2
44. 44. Calculation with Log and Exp Example E: Solve 9*log(2x+1)= 7 Isolate the log-part, log(2x+1) = 7/9 Write it in exp-form 2x + 1 = 107/9 Save for x: 2x = 107/9 – 1 x = (107/9 – 1)/2  2.50
45. 45. Calculation with Log and Exp Example E: Solve 9*log(2x+1)= 7 Isolate the log-part, log(2x+1) = 7/9 Write it in exp-form 2x + 1 = 107/9 Save for x: 2x = 107/9 – 1 x = (107/9 – 1)/2  2.50 Example F: Solve 2.3*log(2–3x)+4.1 = 12.5
46. 46. Calculation with Log and Exp Example E: Solve 9*log(2x+1)= 7 Isolate the log-part, log(2x+1) = 7/9 Write it in exp-form 2x + 1 = 107/9 Save for x: 2x = 107/9 – 1 x = (107/9 – 1)/2  2.50 Example F: Solve 2.3*log(2–3x)+4.1 = 12.5 2.3*log(2–3x) + 4.1 = 12.5
47. 47. Calculation with Log and Exp Example E: Solve 9*log(2x+1)= 7 Isolate the log-part, log(2x+1) = 7/9 Write it in exp-form 2x + 1 = 107/9 Save for x: 2x = 107/9 – 1 x = (107/9 – 1)/2  2.50 Example F: Solve 2.3*log(2–3x)+4.1 = 12.5 2.3*log(2–3x) + 4.1 = 12.5 2.3*log(2–3x) = 12.5 – 4.1 2.3*log(2–3x) = 8.4
48. 48. Calculation with Log and Exp Example E: Solve 9*log(2x+1)= 7 Isolate the log-part, log(2x+1) = 7/9 Write it in exp-form 2x + 1 = 107/9 Save for x: 2x = 107/9 – 1 x = (107/9 – 1)/2  2.50 Example F: Solve 2.3*log(2–3x)+4.1 = 12.5 2.3*log(2–3x) + 4.1 = 12.5 2.3*log(2–3x) = 12.5 – 4.1 2.3*log(2–3x) = 8.4 log(2 – 3x) = 8.4/2.3
49. 49. Calculation with Log and Exp Example E: Solve 9*log(2x+1)= 7 Isolate the log-part, log(2x+1) = 7/9 Write it in exp-form 2x + 1 = 107/9 Save for x: 2x = 107/9 – 1 x = (107/9 – 1)/2  2.50 Example F: Solve 2.3*log(2–3x)+4.1 = 12.5 2.3*log(2–3x) + 4.1 = 12.5 2.3*log(2–3x) = 12.5 – 4.1 2.3*log(2–3x) = 8.4 log(2 – 3x) = 8.4/2.3 2 – 3x = 108.4/2.3
50. 50. Calculation with Log and Exp Example E: Solve 9*log(2x+1)= 7 Isolate the log-part, log(2x+1) = 7/9 Write it in exp-form 2x + 1 = 107/9 Save for x: 2x = 107/9 – 1 x = (107/9 – 1)/2  2.50 Example F: Solve 2.3*log(2–3x)+4.1 = 12.5 2.3*log(2–3x) + 4.1 = 12.5 2.3*log(2–3x) = 12.5 – 4.1 2.3*log(2–3x) = 8.4 log(2 – 3x) = 8.4/2.3 2 – 3x = 108.4/2.3 2 – 108.4/2.3 = 3x
51. 51. Calculation with Log and Exp Example E: Solve 9*log(2x+1)= 7 Isolate the log-part, log(2x+1) = 7/9 Write it in exp-form 2x + 1 = 107/9 Save for x: 2x = 107/9 – 1 x = (107/9 – 1)/2  2.50 Example F: Solve 2.3*log(2–3x)+4.1 = 12.5 2.3*log(2–3x) + 4.1 = 12.5 2.3*log(2–3x) = 12.5 – 4.1 2.3*log(2–3x) = 8.4 log(2 – 3x) = 8.4/2.3 2 – 3x = 108.4/2.3 2 – 108.4/2.3 = 3x 2 – 108.4/2.3 = x  -14953
52. 52. Solve the following exponential equations, give the exact and the approximate solutions. 1. 5e2x = 7 2. 3e - 2x+1 = 6 Exact answer: x = ½* LN(7/5) Exact answer: x = (1 – LN(2)) /2 Aproxímate: 0.168 Aproxímate: 0.153 3. 4 – e 3x+ 1 = 2 4. 2* 10 3x - 2 = 5 Exact answer: x = (LN(2) – 1)/3 Exact answer: x = (LOG(5/2) + 2)/3 Approximate: - 0.102 Approximate: 0.799 5. 6 + 3* 10 1- x = 10 6. -7 – 3*10 2x - 1 = -24 Exact answer: x = 1 – LOG(4/3) Exact answer: x = (LOG(17/3)+1)/2 Aproxímate: 0.875 Aproxímate: 0.877 7. 8 = 12 – 2e 2- x 8. 5*10 2 - 3x + 3 = 14 Exact answer: x = 2 – LN(2) Exact answer: x = (2 – LOG(11/5)) /3 Approximate: 1.31 Approximate: 0.553 Solve the following log equations, give the exact and the approximate solutions. 9. LOG(3x+1) = 3/5 10. LN(2 – x) = -2/3 Exact answer: x = (103/5 – 1)/3 Exact answer: x = 2 – e -2/3 Approximate: 0.994 Approximate: 1.49 11. 2LOG(2x –3) = 1/3 12. 2 + Log(4 – 2x) = -8 Exact answer: x = (101/6 + 3)/2 Exact answer: x = (4 – 10-10)/2 Approximate: 2.23 Approximate: 2.000 13. 3 – 5LN(3x +1) = -8 14. -3 +5LOG(1 – 2x) = 9 Exact answer: x = (e11/5 – 1 )/3 Exact answer: x = (1 – 10 12/5)/2 Approximate: 2.68 Approximate: -125 15. 2LN(2x – 1) – 3 = 5 16. 7 – 2LN(12x+15) =23 Exact answer: x = (e4+1)/2 Exact answer: x = (e-8 – 15 )/12 Approximate: 27.8 Approximate: -1.25