Algebra va matematik analiz

4-misol. a) ln 10 = = =2,30259...;

lge = = = 0,43429...

5-misol. a) lg1000⁶⁷ ; b) lne^4,8 larni hisoblang.
Yechish: a) lg l000⁶⁷ = lg10^3*67 = lg 10²⁰¹ = 201*lg10 = 201*1=201;
b) lne^4,8 = 4,81ne = 4,8*1=4,8.
6-misol. Jadvalda lg3 = 0,4771 ekanligi berilgan.
a) lg270 ni; b) 3¹⁰⁰⁰ ni toping.
Yechish: a) lg270 = lg3³*10 = 31g3 + lg10 = 3*0,4771 + 1 = 2,4313.
b) 3¹⁰⁰⁰ = x deb, bu tenglikni logarifmlasak, lgx =1000 lg3= 477,1 yoki bundan x=10^477,1 hosil bo'ladi.
Demak, 3¹⁰⁰⁰ = 10^477,1 =1000...0. 477 ta


Mashqlar:
1. a>0, a 1 bo'lsa, ifodaning qiymatini toping:
a) log_a³a; b) log _a⁴ a^1/3 ; d) log_1/a a⁷;
e) log ; f) log_a^-1 , g) log_a² a^-5.
2. x ni toping:
a) log_0,1 x = -2; b) log₃₆ x =1/2; d) log_x 9 = -l; e) log 8 = 3.
4. Hisoblang:
a) log 81; b) log₁₆ ; d) log_0,001 ;
e) log 1/64 ; f) (log₂log₄log₈16)*10^{1/2 lg4-lg 2+lg 0,1} ;
g) log₂3*log₃4*log₄5*log₅6*log₆5*log₅4*log₄3*log₃2; j) lg6; k) lg72.
5. Agar: a) log₆8 = c bo'lsa, log₂₄72; b) log₃₆8 = b bo'lsa, log₃₆9;
d) log₁₀₀₀9 = a va log₁₀₀₀4 = b bo'lsa, log₅6 ni toping.
6. Tengsizlik a ning qanday qiymatlarida o'rinli:
a) log₅a < log₅3a; b) log_0,6a > log_0,6 a/2; d) log_a a2,2 ?
7.Funksiyalarning va ularga teskari funksiyalarning aniqlanish sohalarini toping:
a) y =lg(x² + 6x); b) y = lg(10^3x +3); d) y = log₂ (x-8) + log₂ (8-x).


4. Ko'rsatkichli va logarifmik ifodalarni ayniy almashtirishlar.
Oldingi bandlarda logarifmning va logarifmik funksiyaning, shuningdek, darajaning va ko'rsatkichli funksiyaning xossalari bilan tanishgan edik. Bu xossalardan logarif­mik va ko'rsatkichli ifodalarni shakl almashtirishlarda foydalaniladi.
7-miso1. 3^2+log₃² ni hisoblang.
Yechish. 3^2+log₃² = 3² *3^log₃² = 9*2 = 18.
8-misol. a^log_b^c =c^log_b^a (a>0, a1, b>0, b0, c> 0) tenglikni isbotlang.
Isbot: Logarifmning log_ab^p = p*log_ab (a > 0, a 1, b>0, pR) xossasidan foydalansak, log_ba*log_bc = log_ba*log_bc tenglikdan log_b(a^log_b^c) = log_b(c^log_b^a) teng­likni hosil qilamiz. Logarifmik funksiyaning monotonlik xossasidan a^log_b^c = c^log_b^a ekanligi kelib chiqadi.
9-miso1. A = log₄ x²/4 - 2log₄(4x⁴) ifodani soddalashtiring va uning x = -2 dagi qiymatini toping.
Yechish. Log_ab²ⁿ = 2nlog|b| (a>0, a1, b0, nN) bo'lgani uchun log₄ x²/4 =log₄x² - log₄4 = 2log₄|x| - 1 va log₄ (4x⁴) = log₄4 + log₄ x⁴ = 1 + 41og₄|x| tengliklar o'rinli.
U holda, A = 2log₄ x| - 1 - 2(1 + 4log₄|x|) = -3 - 61og₄|x|. x = -2 bo'lsa,
A = -3 - 61og₄|- 2 = -3 – 6log₄2 = -6.
10-misol. y = log (x-1/2) + log₂ funksiyaning grafigini yasang.
Y echish. Funksiya ifodasini soddalashtirmay, grafikni yasashga harakat qilish maqsadga muvofiq emas ekan­ligi ko'rinib turibdi. Shu sababli dastlab funksiyaning ifodasini soddalashtiramiz:
Log₂ = log₂ = log₂|2x-1| = log₂(2 • x –1/2)= 1 + log₂ x –1/2 tenglik o'rinlidir. Bu yerda funksiyaning aniqlanish sohasi |1/2; + | oraliqdan iboratligini ko'ramiz.
x > ½ da esa log (x-1/2) = - log₂(x–1/2) bo'lgani uchun

(4- rasm)

y = log (x-1/2)+log₂ = - log₂(x–1/2) + (l + log₂ x -1) = -log₂ (x –1/2) + 1 + log₂ x –1/2 = 1 ga ega bo'lamiz.
Endi funksiya grafigini yasash (4- rasm) qiyinchilik tug'dirmaydi.


Mashqlar :
1. Ifodani soddalashtiring:
a) ;
b) 81^1/log₉³ + 27^log₉³⁶ + 3^4/log₇⁹ ;
2. x ni toping:
a log₃ x = 2 log₃ (a + b)- 2/3log₃ (a -b) +1/2log₃a;
b log₄ x = log₄ (a - b) + 1/3 (2 log₄ a + 3 log₄ b);
d) log₅x = 51og₅m +1/2[ log₅(m+n)+1/3 log₅ (m-n)-log₅m –log₅n);
e log₆ x = - log₆ (a + b) +2/5( 2log₆ a +1/2 log₆ b – 1/3(log₆a-log₆b)-log₆a).
3. Sonning musbat yoki manfiy ekanini aniqlang:
a) lg2 + lg3 + lg 0,16;
log 7-1/2log l,2-31og 2;


Nazorat savollari:
1.Darajalar ustida amallar?
2.Ko’rsatkichli funksiya deb nimaga aytiladi?
3. Logarifm deb nimaga aytiladi?
4.Logarifmlash va potensirlashni tushuntirib bering?
5.Logarifmik funksiya deb nimaga aytiladi?
6. Asosiy logarifmik ayniyatni yozib bering?
7. O’nli logarifmga ta’rif bering?
8. Natural logarifmga ta’rif bering?
9. Xarakteristika va mantissani izohlang?
10. Logarifmlarni ayniy shakl almashtirishga misol keltiring?