4.4 the logarithm functions t
- 1. The Logarithmic Functions Example B. Rewrite the log-form into the exp-form. a. log3(1/9) = –2 3-2 = 1/9 b. 2w = logv(a – b) v2w = a – b Example A. Rewrite the exp-form into the log-form. a. 42 = 16 log4(16) = 2 b. w = u2+v logu(w) = 2+v To convert the exp-form to the log–form: b = y x logb( y ) = x→ To convert the log–form to the exp–form: b = y x logb( y ) = x→ The output of logb(x), i.e. the exponent in the defined relation, may be positive or negative.
- 2. The Logarithmic Functions Example C. a. Rewrite the exp-form into the log-form. 4–3 = 1/64 8–2 = 1/64 log4(1/64) = –3 log8(1/64) = –2 exp–form log–form b. Rewrite the log-form into the exp-form. (1/2)–2 = 4log1/2(4) = –2 log1/2(8) = –3 exp–formlog–form (1/2)–3 = 8 Example F. Solve for x a. log9(x) = –1 Drop the log and get x = 9–1. So x = 1/9
- 3. b. logx(9) = –2 Drop the log and get 9 = x–2, i.e. 9 = So 9x2 = 1 x2 = 1/9 x = 1/3 or x= –1/3 Since the base b > 0, so x = 1/3 is the only solution. The Common Log and the Natural Log
- 4. The Logarithmic Functions (1, 0) (2, 1) (4, 2) (8, 3) (16, 4) (1/2, -1) (1/4, -2) y=log2(x) Graphs of the Logarithmic Functions 1/4 -2 1/2 -1 1 0 2 1 4 2 8 3 x y=log2(x) Recall that the domain of logb(x) is the set of all x > 0. Hence to make a table to plot the graph of y = log2(x), we only select positive x’s. In particular we select x’s related to base 2 for easy computation of the y’s. x y
- 5. The Logarithmic Functions x y (1, 0) (8, -3) To graph log with base b = ½, we have log1/2(4) = –2, log1/2(8) = –3, log1/2(16) = –4 (4, -2) (16, -4) y = log1/2(x) x x y (1, 0)(1, 0) y = logb(x), b > 1 y = logb(x), 1 > b Here are the general shapes of log–functions. y (b, 1) (b, 1)
- 6. 3x2 y log( ) = log( ), by the quotient rule = log (3x2) – log(y1/2) product rule power rule = log(3) + log(x2) – ½ log(y) = log(3) + 2log(x) – ½ log(y) 3x2 y 3x2 y1/2 Properties of Logarithm a. Write log( ) in terms of log(x) and log(y). log(3) + 2log(x) – ½ log(y) power rule = log(3) + log(x2) – log(y1/2) product rule = log (3x2) – log(y1/2)= log( )3x2 y1/2 b. Combine log(3) + 2log(x) – ½ log(y) into one log. Example G. quotient rule
- 7. we have that: a. logb(expb(x)) = x or logb(bx) = x b. expb(logb(x)) = x or blog (x) = x Properties of Logarithm b Example H. Simplify a. log2(2-5) = -5 b. 8log (xy) = xy c. e2+ln(7) = e2·eln(7) = 7e2 8
- 8. 1. Exercise A. Rewrite the following exp-form into the log-form. 2. 3. 4. 5. 6. 7. 8. 9. 10. The Logarithmic Functions Exercise B. Rewrite the following log–form into the exp-form. 52 = 25 33 = 27 1/25 = 5–2 x3 = y y3 = x ep = a + b e(a + b) = p 10x–y = z11. 12. 1/25 = 5–2 1/27 = 3–3 1/a = b–2 A = e–rt log3(1/9) = –2 –2 = log4(1/16)13. 14. log1/3(9) = –215. 2w = logv(a – b)17. logv(2w) = a – b18.log1/4(16) = –216. log (1/100) = –2 1/2 = log(√10)19. 20. ln(1/e2) = –221. rt = ln(ert)23. ln(1/√e) = –1/224.log (A/B) = 322.
- 9. Exercise C. Convert the following into the exponential form then solve for x. The Logarithmic Functions logx(9) = 2 x = log2(8)1. 2. log3(x) = 23. 5. 6.4. 7. 9.8. logx(x) = 2 2 = log2(x) logx(x + 2) = 2 log1/2(4) = x 4 = log1/2(x) logx(4) = 1/2 11. 12.10. 13. 15.14. ln(x) = 2 2 = log(x) log(4x + 15) = 2 In(x) = –1/2 a = In(2x – 3) log(x2 – 15x) = 2
- 10. Ex. D Disassemble the following log expressions in terms of sums and differences logs as much as possible. Properties of Logarithm 5. log2(8/x4) 6. log (√10xy) y√z3 2. log (x2y3z4) 4. log ( ) x2 1. log (xyz) 7. log (10(x + y)2) 8. ln ( ) √t e2 9. ln ( ) √e t2 10. log (x2 – xy) 11. log (x2 – y2) 12. ln (ex+y) 13. log (1/10y) 14. log ( )x2 – y2 x2 + y2 15. log (√100y2) 3 3. log ( ) z4 x2y3 16. ln( )x2 – 4 √(x + 3)(x + 1) 17. ln( )(x2 + 4)2/3 (x + 3)–2/3(x + 1) –3/4
- 11. E. Assemble the following expressions into one log. Properties of Logarithm 2. log(x) – log(y) + log(z) – log(w) 3. –log(x) + 2log(y) – 3log(z) + 4log(w) 4. –1/2 log(x) –1/3 log(y) + 1/4 log(z) – 1/5 log(w) 1. log(x) + log(y) + log(z) + log(w) 6. –1/2 log(x – 3y) – 1/4 log(z + 5w) 5. log(x + y) + log(z + w) 7. ½ ln(x) – ln(y) + ln(x + y) 8. – ln(x) + 2 ln(y) + ½ ln(x – y) 9. 1 – ln(x) + 2 ln(y) 10. ½ – 2ln(x) + 1/3 ln(y) – ln(x + y)
- 12. Continuous Compound Interest F. Given the following projection of the world populations, find the growth rate between each two consecutive data. Is there a trend in the growth rates used?
- 13. (Answers to the odd problems) Exercise A. Exercise B. 13. 3−2 = 1/9 15. 1 3 −2 = 9 17. 𝑦2𝑤 = 𝑎 − 𝑏 19. 10−2 = 1/100 21. 𝑒−2 = 1/𝑒2 23. 𝑒 𝑟𝑡 = 𝑒 𝑟𝑡 1. 𝑙𝑜𝑔5 25 = 2 3. 𝑙𝑜𝑔3 27 = 3 7. 𝑙𝑜𝑔 𝑦(𝑥) = 3 9. 𝑙𝑜𝑔 𝑒 𝑎 + 𝑏 = 𝑝 5. 𝑙𝑜𝑔391/27) = −3 11. 𝑙𝑜𝑔 𝑒 𝐴 = −𝑟𝑡 Exercise C. 1. 𝑥 = 3 3. 𝑥 = 9 5. 𝑥 = 4 7. 𝑥 = −2 9. 𝑥 = 16 11. 𝑥 = 100 13. 𝑥 = 1 𝑒 15. 𝑥 = −5, 𝑥 = 20 The Logarithmic Functions
- 14. Exercise D. 5. 𝑙𝑜𝑔2 8 − 4𝑙𝑜𝑔2(𝑥) 1. log(𝑥) + log(𝑦) + log(𝑧) 7. log(10) + 2log(𝑥 + 𝑦) 9. 2ln(𝑡) − 1/2ln(𝑒) 11. log(𝑥2 – 𝑦2) 13. log(1) − 𝑦𝑙𝑜𝑔(10) 3. 2log(𝑥) + 3log(𝑦) − 4log(𝑧) 15. 1/3log(100) + 2/3log(𝑦) 17. 2/3ln(𝑥 + 4) + 2/3ln(𝑥 + 3) + 3/4ln(𝑥 + 1) Exercise E. 3. log 𝑦2 𝑤4 𝑥𝑧3 1. log(𝑥𝑦𝑧𝑤) 5. 𝑙𝑜𝑔 (𝑥 + 𝑦)(𝑧 + 𝑤) 7. 𝑙𝑛 𝑥(𝑥+𝑦) 𝑦 9. 𝑙𝑛 𝑒𝑦2 𝑥 Properties of Logarithm