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4.2.1. Logaritmien laskusäännöt

4.2.1. Logaritmien laskusäännöt. Logaritmin laskusääntöjä Olkoon a, b > 0 ja k > 0, k ≠ 1 log k ab= log k a + log k b log k (a/b)= log k a - log k b log k a n = n · log k a E.9. a) lg2 + lg5 = lg (2  5) = lg10 = 1 b) lga 2 + lg(1/a) (a>0) = 2lga + lg1 - lga

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4.2.1. Logaritmien laskusäännöt

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  1. 4.2.1. Logaritmien laskusäännöt

  2. Logaritmin laskusääntöjä Olkoon a, b > 0 ja k > 0, k ≠ 1 logkab= logka + logkb logk(a/b)= logka - logkb logk an = n · logka E.9. a) lg2 + lg5 = lg (2  5) = lg10 = 1 b) lga2 + lg(1/a) (a>0) = 2lga + lg1 - lga =2lga + 0 - lga = lga c) Esitä lg3:n avulla lg3000 = lg(3  1000) = lg3 + lg 1000 = lg3 + 3

  3. E.10. Esitä yhtenä logaritmina • lna + ln6 b) ½ln16 + 1 c) 2ln4 – 3ln2 • a) lna + ln6 = ln6a • b) ½ln16 + 1 • = ln16½ + lne • = ln4 + lne = ln4e • c) 2ln4 – 3ln2 • = ln42 – ln23 • = ln16 – ln8 • = ln (16/8) • = ln2

  4. E.11. (t. 203a,c) • Olkoon a = logk2 ja b=logk3. Esitä a:n ja b:n avulla • logk (3/2) • = logk 3 – logk2 • = b – a • b) logk12 • = logk (4 · 3) • = logk 4 + logk3 = logk22 + logk3 • = 2logk2 + logk3 = 2a + b

  5. Logaritmin kantaluvun vaihtaminen

  6. 4.2.2. Eksponenttiyhtälöitä E.12. 2x = 7 lg2x = lg7 xlg2 = lg7 x = lg7 /lg2  2,807 tai ks. kantaluvun vaihto edellä

  7. E.13. (t. 210a) 3x+1 = 20 lg3x+1 = lg20 (x+1)lg3 = lg20 x  1,72

  8. E.14. Kuinka monessa vuodessa talletuksen arvo viisinkertaistuu, jos vuotuinen korko on 8% ? a = talletuksen arvo alussa 1,08x a = 5a | :a 1,08x = 5 lg 1,08x = lg 5 x  lg 1,08 = lg 5 x = lg 5/ lg1,08 =20,9 V: 21 vuodessa

  9. E.15. (t. 221b) 5x = 17  4x c  12,7

  10. 4.2.3. Logaritmiyhtälöitä logku = logkv  u = v (u, v > 0) E.16. (t. 212a) a) Ratkaise yhtälö log2 (3x – 1) = 1 Määrittelyehto: 3x – 1 > 0 eli x > 1/3 2 1 = 3x – 1 3x = 3 x = 1

  11. b) lgx + lg(x - 3) = 1 (x > 3) lgx(x-3) = lg 10 x(x - 3) = 10 x2 - 3x - 10 = 0, ratkaisukaavalla ( x = -2), x = 5

  12. E.17. 2lnx = 1 Mj: x>0 lnx = ½ x = e½ x = √e E.18. ln3 = -ln(1 – x) Mj: x < 1 ln3 = ln(1 – x)-1 3 = (1 – x)-1 3(1- x) = 1 1 - x = 1/3 x = 2/3

  13. E.19. (t. 223a) Määritelty, kun x > 0 x = 4

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