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Logarifmik funksiyalarning xossalari va grafigi
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Logarifmik funksiyalarning xossalari va grafigi Logarifmik funksiyaning ko'rinishi: ( ) log , 0, 1, 0 a y x a a x = > ¹ > . 1. Aniqlanish sohasi: ( ) ( ) 0;
= + ¥
barcha musbat sonlar to'plami. 2. Qiymatlar sohasi: ( )
; E y = -¥ + ¥
barcha haqiqiy sonlar to'plami. 3. Logarifmik funksiya aniqlanish sohasida agar 1
> bo'lsa, o'suvchi. Agar 0 1 a < < bo'lganda kamayuvchi. 4. Agar 1
> bo'lsa, logarifmik funksiya 1
> da musbat 0 1 x < < da esa manfiy qiymatlar qabul qiladi. 5. Agar 0
a < < bo'lsa, logarifmik funksiya 0 1
qiymatlar, 1
> da esa manfiy qiymatlar qabul qiladi. 6. log
a y x = logarifmik funksiya juft ham, toq ham, davriy ham emas. 7. Logarifmik funksiyaning grafigi (1; 0)
nuqtadan o’tadi. Click here to buy A B B Y Y PD F Transfo rm er 2 .0 w w w .A B B Y Y. c o m Click here to buy A B B Y Y PD F Transfo rm er 2 .0 w w w .A B B Y Y. c o m 43 8. ( ) log
, 0, 1, 0 a y x a a x = > ¹ > funksiyaning grafigi: ( ) ( ) 0; D y = + ¥ , ( ) ( ) ;
= -¥ + ¥ .
Ushbu (
log 0, 1,
a x b a a b R = > ¹ Î ko`rinishdagi tenglamalarga sodda logarifmik tenglama diyiladi. Yechishda qo’llaniladigan asosiy ekvivalent almashtirishlar: 1. log , 0 (
1, 0)
a x b x a x a a = Û =
> ¹ > . 2. log ( ) ( )
, ( )
0, b a f x b f x a f x b R = Û
= > Î ( 1, 0) a a ¹ > . 3. ( ) ( ) 0, ( ) 0, ( )
1, log
( ) ( )
( ). x b f x x x f x b f x x j j j j > > ¹ ìï = Û í = ïî 4. ( )
( ) 0 , 0 , 1, lo g ( )
( ) ( )
. a x f x a a f x x f x a j j > > ¹ ìï = Û í = ïî 5. ( ) 0,
( ) 0, 0, 1, log ( )
log ( )
( ) ( ).
a a f x g x a a f x g x f x g x > > > ¹ ì = Û í
= î 6. ( ) ( )
( ) 0, ( ) 0,
log log
( ) 1, 0, ( ) 1, 0, ( ) ( );
( ) ( ).
f x g x f x g x A A f x A yoki g x A f x g x f x g x > > ì ì ï ï = Û ¹ > ¹ > í í ï ï = = î î 7. og ( ) l ( )
0, ( )
0, ( )
0, 1, ( ) ( ). g x a f x g x f x a a a f x g x > > ì ï = Û > ¹ í ï = î 8. ( ) ( )
( ) 0, ( ) 0, log
( ) log
( ) log
( ) 0, 1 ( ) ( ) .
a a f x g x f x g x m x a a f x g x m x > > ìï + = > ¹ Û í × = ïî Click here to buy A B B Y Y PD F Transfo rm er 2 .0 w w w .A B B Y Y. c o m Click here to buy A B B Y Y PD F Transfo rm er 2 .0 w w w .A B B Y Y. c o m 44 9. ( ) ( ) 2 1 ( ) 0, 2 1 log ( ) log
( )
0, 1, ( )
( ). a a n g x n f x g x a a n N f x g x + > ìï + = > ¹ Î Û í = ïî 10. ( ) 2 ( )
0, 2 log
( ) log
( ) 0, 1, ( ) ( ). a a n f x n f x g x a a n N f x g x > ìï = > ¹ Î Û í
= ïî 11. (log ) 0, 0, 1 log ,
( ) 0.
a f x a a x t f t = > ¹ Û = = 12. log
log log
, 0, 0, 0, 1, 1, 1, 0 a b c x x x d a b c a b c x + + = > > > ¹ ¹ ¹ > Û
log log
log . log log a a a a a x x x d b c Û + + =
Logarifmik tengsizliklar ushbu ekvivalent almashtirish yordamida yechiladi: 1. 0 1, lo g
( ) ( )
0 , ( )
. a b a f x b f x f x a ì < <
ï ³ Û > í ï £ î 2. 1, lo g
( ) ( )
0, ( )
. a b a f x b f x f x a ì >
ï ³ Û > í ï ³ î 3. 0 1, ( ) 0, 1, ( ) 0, log
( ) log
( ) ( )
0, ( )
0, ( )
( ); ( )
( ). a a a g x a g x f x g x f x f x f x g x f x g x < < > > > ì ì ï ï
Û >
í í ï ï >
î î
4. [ ] [ ] ( ) 0 ( )
1, ( )
1, lo g
( ) ( )
0, ( )
0, ( )
( ) ; ( ) ( ) .
a a f x f x g x a g x g x g x f x g x f x ì ì < < > ï ï ï ï < Û > > í í ï ï > < ï ï î î U 5. ( )
0 ( )
1, ( )
1, lo g
( ) 0 0 ( ) 1 ( ) 1 . f x f x f x g x g x g x < < > ì ì > Û í í < < > î î U 6. ( ) 0 ( ) 1, ( )
1, lo g
( ) 0 ( ) 1 0 ( ) 1 . f x f x f x g x g x g x < < > ì ì < Û í
í >
< î î U 7. ( ) 0 ( )
1, ( )
1, lo g
( ) 0 0 ( ) 1 ( ) 1 . f x f x f x g x g x g x < < > ì ì ³ Û í í < £ ³ î î U 8. ( )
0 ( )
1, ( )
1, lo g
( ) 0 ( ) 1 0 ( ) 1 . f x f x f x g x g x g x < < > ì ì £ Û í í ³
£ î
U 9. ( ) ( ) ( )
1, ( )
0 , lo g
( ) lo g
( ) ( )
0 , 0 ( ) 1, ( )
( ); ( )
( ). x x x f x f x g x g x x f x g x f x g x j j j j > > ì ì ï ï > Û >
< í í ï ï > < î î U Click here to buy A B B Y Y PD F Transfo rm er 2 .0 w w w .A B B Y Y. c o m Click here to buy A B B Y Y PD F Transfo rm er 2 .0 w w w .A B B Y Y. c o m 45 10.
( ) ( ) ( ) 1, ( ) 0 , lo g
( ) lo g
( ) ( )
0 , 0 ( ) 1, ( )
( ) ; ( )
( ) . x x x g x f x g x f x x f x g x f x g x j j j j > > ì ì ï ï £ Û >
< í í ï ï £ ³ î î U T R I G O N O M E T R I Y A Boshlang’ich tushunchalar 1. 0 a -gradusdan radianga o’tish: 1 8 0
p a a = × o o . 2. rad a -radiandan gradusga o’tish: 1 8 0 r a d a a p = × o o . 3. Ta`riflar: 1)
y sin y r a = = ; 2) x cos x r a = = ; 3) , 0 y tg x x a = ¹ ; 4)
, 0
ctg y y a =
¹ ; 5) sin cos
tg a a a = ; 6) cos sin
ctg a a a = . Trigonometrik funksiyalar qiymatlari jadvali Funksiyalar Burchak α, gradus(radian) sin α cos α
tg α ctg α
0° (0) 0 1 0 Mavjud emas 15° (π/12) 3 1
2 2 - 3 1 2 2 + 2 3 - 2 3 + 18° (π /10) 5 1 4 - 5 5 2 2 + 5 1
10 2 5 - + 10 2 5 5 1
+ - 22,5° (π /8) 2 2 2
- 2 2 2 + 2 1
- 2 1
+ 30° (π /6) 1 2 3 2
1 3 3 36° (π /5) 5 5 2 2 - 5 1 4 + 10 2 5 5 1 - + 5 1 10 2 5
+ - 45° (π /4) 2 2 2 2
1 1 60° (π /3) 3 2 1 2
3 1 3 90° (π /2) 1 0 Mavjud emas
0 Click here to buy A B B Y Y PD F Transfo rm er 2 .0 w w w .A B B Y Y. c o m Click here to buy A B B Y Y PD F Transfo rm er 2 .0 w w w .A B B Y Y. c o m 46 75° (5 π /12) 3 1 2 2
+ 3 1
2 2 - 2 3 + 2 3 - 180° (π) 0 -1 0 Mavjud emas 270° (3 π /2) -1 0
emas 0 360° (2 π) 0 1 0 Mavjud emas Trigonometrik funksiyalarning ishoralari Asosiy trigonometrik ayniyatlar 1. 2 2 cos
sin 1 a a + = . 2. ( ) sin 1 ; 2 1 ,
cos 2
n n Z ctg a p a a a a = = ¹ + Î . 3. 1 tg ctg a a × = . 4. cos 1
, sin
ctg n n Z tg a a a p a a = = ¹ Î . 5. 2 2 1 1 cos
tg a a + = . 6. 2 2 1 1 ; , sin ctg n n Z a a p a + = ¹ Î . Download 0.8 Mb. Do'stlaringiz bilan baham: 
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