Algebra va matematik analiz

1-misol. a) log₃x = 9; b) log_x64= 2 tenglamalarni yechamiz.
Yechish. a) Tenglamani potensirlaymiz. Natijada: x=3⁹; b) tenglamani potensirlaymiz: x² = 64, bundan x= 8.
2-misol. a) log₃x < 9; b) log_1/3 x < 9 tengsizliklarni yechamiz.
Yechish. a) oldingi misolda log₃x=9 tenglamaning x = 3⁹ ildizi topilgan edi. Asos a = 3>l, b = 9. Yechim: (0; 3⁹) yoki 09;
b) a = 1/3 (0; 1) bo'lgani uchun yechim (3^-9; +) oraliqdan iborat.
1-teorema. Log_a f(x) = log_a gx) (a> 0, a1) tenglama
{fx=gx,
{fx>0 1 sistemaga teng kuchlidir.
Isbot. y=log_at (a>0, a1) logarifmik funksiya monoton. Shunga ko'ra log_af(x) = log_ag(x) tengligining bajarilishi uchun f(x) = g(x) bo'lishi kerak. Demak, f(x) > 0 bo'lganda log_a f(x) = log_ag(x) tenglama f(x) = g(x) tenglamaga teng kuchli.
1^/-teorema. Log_a f(x) = log_a gx) (a> 0, a1) tenglama
{fx=gx,
{gx>0 1^/ sistemaga teng kuchlidir.
Bu teoremani isbotlashda 1- teoremaning isbotidagi kabi mulohazalar yuritiladi
2-teorema. Agar 0 < a < 1 bo'lsa, log_a fx) >log_ag(x) tengsizlik 0l bo'lsa, f(x)>g(x) >0 qo'sh tengsizlikka teng kuchlidir.
Bu teoremaning isboti logarifmik funksiyaning monotonligidan kelib chiqadi.
3-misol. tenglamani yechamiz.
Yechish. 1) Tenglamaning aniqlanish sohasini topamiz:
{x + 7 > 0, {x>-7,
{x-5 > 0, {x>5, {x>5,
{lg8-lgx-50 {x-58 {x>13;
2) ifodani sodda ko'rinishga keltirish maqsadida ayniy almashtirishlarni bajaramiz:
lg - lg2 = lg(x - 5) - lg8 lg /2 = lgx-5/8 => /2=x-5/8  ² = x-5/4² x²-26x+87=0.
Bundan x = 29 ekani aniqlanadi.
Mashqlar:
1. Tenglamani yeching:
a) 2-ln(x - 3) = lnx - ln4; b) x^lgx = x¹⁰; d) 0,l*x^lgx-4= 100³;
e) log₂₅ x² - 10x+ 9) = 2; f) 2log_x x⁴+log₂ x = 4; e log₃(3^x - 8) = 2 - x;
f log₄(2log₃(l + log₂(l + 3log₃x))) =1/2; g) x ^{log x} =9; h log₂(3 ^x + 4)=2-5 ^x ;
2. Tengsizlikni yeching:
a) lg²x²+5lgx>-l,25; b) log_x( -x-l)l; d) (log_x2)(log_2xt2)(log₂4x) > 1;
e) log_x(24 - 2x - x²) < 1; f) log_1/3x+5x-6 >2; g) (log_2xt0,5)²log_2x(2x²);
h 2 ^xlog₃x=log₃x2^x+1+2; i log_xlog₉3^x-9<1; j log 2+x<1;