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Chapter 4

Chapter 4. Logarithms. In a plentiful springtime, a population of 1000 mice will double every week. The population after t weeks is given by the exponential function P(t ) = 1000 x 2 t mice . Things to think about: a. What does the graph of the population over time look like ?

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Chapter 4

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  1. Chapter 4 Logarithms

  2. In a plentiful springtime, a population of 1000 mice will double every week. The population after t weeks is given by the exponential function P(t) = 1000 x 2tmice. Things to think about: a. What does the graph of the population over time look like? b. How long will it take for the population to reach 20 000 mice? c. Can we write a function for t in terms of P, which determines the time at which the population P is reached? d. What does the graph of this function look like?

  3. Logarithms in base 10 If f(x) = 10x then f-1(x) = log10 x. y = 10x y = x y = log10 x

  4. The logarithm in base 10 of a positive number is the exponent when the number is written as a power of 10.

  5. a = 10log a for any a > 0 log 10x = x for any x єR

  6. Exercise 4A 1. Without using a calculator, find: b. log 0.001 e. log j. log

  7. 2. Simplify a. log 10n c. log

  8. 3. Use your calculator to write the following in the form 10x where x is correct to 4 decimal places. b. 60 i. 0.15 g. 1500

  9. 4a. Use your calculator to find: i. log 3 ii. log 300 4b. Explain why log 300 = log 3 + 2.

  10. 6. Find x if: a. log x = 1 e. log x = ½ i. log x ≈ 0.8351 l. log x ≈ -3.1997

  11. Logarithms in base a f(x) = ax If f(x) = ax then f-1(x) = loga x. (1, a) f -1(x) = loga x (a, 1) If b = ax , a ≠1, a > 0, we say that x is the logarithm in base a, of b, and that b = ax x = logab, b > 0.

  12. provided x > 0

  13. Exercise 4B 1. Write an equivalent exponential equation for: c. log10 (0.1) = -1 e. log2 8 = 3 i.

  14. 2. Write an equivalent logarithmic equation for: a. 43 = 64 b. c. 10-2 = 0.01

  15. 3. Find: b. log10 (0.01) i. log2 0.125 k. log4 16

  16. 5. Solve for x: • log2 x = 3 • log4 x = ½ • logx 81 = 4 • log2 (x – 6) = 3

  17. Laws of Logarithms If A and B are both positive, then • log A + log B = log (AB) • log A – log B = log (A/B) • nlog A = log(An) If A and B are both positive and c ≠ 1 and c > 0, then 4. logc A + logc B = logc(AB) 5. logcA – logcB = logc(A/B) 6. nlogcA = logc(An)

  18. Exercise 4c.1 1 Write as a single logarithm or as an integer: b. log 4 + log 5 d. log p – log m h. 1 + log23 n. log3 6 –log3 2 –log33 r. log (4/3) + log 3 + log 7

  19. 2. Write as a single logarithm or integer: b. 2 log 3 + 3 log 2 d. 2 log3 5 –3 log32 f. 1/3 log (1/8) h. 1 –3 log 2 + log 20

  20. 3. Simplify without using a calculator: b. e.

  21. 4. Show that: a. log 9 = 2 log 3 e. log 5 = 1 –log 2

  22. 5. If p = logb 2, q = logb 3, and r = logb 5 write in terms of p, q,andr: a. logb 6 e. logb(5/32)

  23. 7. If logtM = 1.29 and logt N2 = 1.72 find: a. logtN b. logt(MN) c.

  24. Exercise 4c.2 1. Write the following as logarithmic equations (in base 10), assuming all terms are positive: b. y = 20b3 f. k. S = 200 · 2t

  25. 2. Write the following equations without logarithms: a. log D = log e + log 2 d. lognM = 2 logn b + lognc h. logaQ = 2 – logax

  26. Natural Logarithms y = ex ln ex = x and elnx = x. y = x (1, e) ax = ex ln a, a > 0 y = ln x (e, 1)

  27. Exercise 4D.1 1. Without using a calculator find: b. ln e3 f. h.

  28. 2. Simplify: b. e 2 ln3 c. e -ln 5 d. e -2 ln2

  29. 3. Explain why ln(-2) and ln 0 cannot be found. 4. Simplify: b. ln(e • ea) c. ln(ea•eb) e. ln(ea/eb)

  30. 5 Use your calculator to write the following in the form ek where k is correct to 4 decimal places: c. 6000 e. 0.006

  31. 6. Find x if: a. ln x = 3 f. ln x ≈ 0.835 h. ln x ≈ -3.2971

  32. Laws of Natural Logarithms For positive A and B: lnA+ lnB = ln(AB) lnA – lnB = ln(A/B) n lnA = ln(An)

  33. Exercise 4D.2 1. Write as a single logarithm or integer: a. ln15 + ln3 c. ln 20 – ln 5 h. ln6 – 1 l. ln 12 –ln 4 –ln 3

  34. 2. Write in the form ln a, a єQ: b. 3 ln 2 + 2 ln5 f. 1/3 ln( 1/27 ) i. -2 ln(1/4)

  35. 5 Write the following equations without logarithms, assuming all terms are positive: a. lnD = lnx +1 e. lnB = 3lnt – 1 h. lnD ≈ 0.4 lnn– 0.6582

  36. Exponential equation using logarithm 1. Solve for x, giving an exact answer in base 10: b. 3x = 20 e. f. 10x= 0.00001

  37. 2. Solve for x, giving an exact answer: a. ex = 10 c. 2ex = 0.3 f.

  38. 3 Consider the equation R = 200 ( 2) 0.25t. a. Rearrange the equation to give t in terms of R. b. Hence find t when: i. R = 600 ii. R = 1425

  39. 5. Solve for x, giving an exact answer: a. 4 • 2-x= 0.12 c. 32 • 3-0.25x= 4 f. 41e0.3x – 27 = 0

  40. 6. Solve for x: a. e2x= 2ex c. e2x– 5ex+6 = 0 e. 1 +12e-x= ex

  41. 7. Find algebraically the point(s) of intersection of: a. y = ex and y = e2x– 6 b. y = 2ex + 1 and y = 7 –ex c. y = 3 –ex and y = 5e-x–3

  42. The change of base rule For a, b, c > 0 and b, c ≠ 1.

  43. Exercise 4F • Use the rule to find, correct to 3 significant figures: • log3 12 c. log3(0.067) • log1/2 1250

  44. 2. Use the rule to solve, correct to 3 significant figures: a. 2x= 0.051 c. 32x+1 = 4.069

  45. Solve for x exactly: a. 25x–3(5x) = 0 c. 2x–2(4x) = 0

  46. 4. Solve for x exactly: log4 x3 + log2√x = 8

  47. Graphs of Logarithmic Function If f(x) = ax where a > 0, a ≠1, then f -1(x) = loga x.

  48. y = axfor a > 1 y = axfor 0 < a < 1 y = loga x fora > 1 y = loga x for 0 < a < 1:

  49. In general, y = loga(g(x)) is defined when g(x) > 0.

  50. Exercise 4G 1. For the following functions f: i. Find the domain and range. ii. Find any asymptotes and axes intercepts. iii. Sketch the graph of y = f(x) showing all important features. iv. Solve f(x) =-1 algebraically and check the solution on your graph. v. Find f-1 and explain how to verify your answer.

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