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Logarifm ta’rifi va xossalari
maktab darsligi: matematika 10-sinf, logarifm va uning xossalari
Maqsad: Logarifmni ta’rifdan hisoblash va xossalar bilan ifodani soddalashtirish.
Qoida va g‘oya
Logarifm: log_a b = c degani a^c = b, bunda a > 0, a ≠ 1, b > 0. Ya’ni logarifm — b ni hosil qilish uchun a ni qanday darajaga ko‘tarish kerakligini ko‘rsatadi. Asosiy ayniyat: a^(log_a b) = b. Maxsus qiymatlar: log_a 1 = 0, log_a a = 1. Xossalar (x, y > 0): log_a (xy) = log_a x + log_a y; log_a (x/y) = log_a x - log_a y; log_a (x^k) = k · log_a x; log_(a^k) b = (1/k) · log_a b. O‘nli logarifm lg b = log_10 b, natural logarifm ln b = log_e b (e ≈ 2,718). Logarifm belgisi ostida son doim musbat, asos 1 ga teng bo‘lmagan musbat son.
Namunaviy masalalar
log_2 32 va log_3 (1/9) ni hisoblang.
Yechim: 2^5 = 32, demak log_2 32 = 5. 3^(-2) = 1/9, demak log_3 (1/9) = -2.
Javob: 5 va -2
Yechim: 2^5 = 32, demak log_2 32 = 5. 3^(-2) = 1/9, demak log_3 (1/9) = -2.
Javob: 5 va -2
5^(log_5 7) ni hisoblang.
Yechim: Asosiy ayniyat bo‘yicha 7.
Javob: 7
Yechim: Asosiy ayniyat bo‘yicha 7.
Javob: 7
log_2 12 + log_2 (4/3) ni hisoblang.
Yechim: log_2 (12 · 4/3) = log_2 16 = 4.
Javob: 4
Yechim: log_2 (12 · 4/3) = log_2 16 = 4.
Javob: 4
Tipik xatolar
- log_a (x + y) ni log_a x + log_a y deb yozish: yig‘indining logarifmi ajratilmaydi.
- Logarifm belgisi ostida manfiy yoki nol songa yo‘l qo‘yish.
- log_a (x^k) = (log_a x)^k deb yozish: k ko‘paytuvchi bo‘lib chiqadi.
Mashq
1
log_4 64 ni hisoblang.
3
2
log_(1/2) 8 ni hisoblang.
-3
3
10^(lg 3) ni hisoblang.
3
4
log_6 4 + log_6 9 ni hisoblang.
2