Logarifm va uning xossalari

Ta'rif

b sonning a asosli logarifmi deb, b ni hosil qilish uchun a ni ko'tarish kerak bo'lgan darajaga aytiladi: logₐb = c ⇔ aᶜ = b.

• Mavjudlik sharti: a > 0, a ≠ 1, b > 0.
• Maxsus belgilar: lg b = log₁₀b (o'nli), ln b = logₑb (natural).
• log₂8 = 3, chunki 2³ = 8. logₐ1 = 0, logₐa = 1.

Asosiy xossalar

Xossa	Formula
Ko'paytma	logₐ(bc) = logₐb + logₐc
Bo'linma	logₐ(b/c) = logₐb − logₐc
Daraja	logₐ(bⁿ) = n · logₐb
Asosdagi daraja	b ning aᵏ asosli logarifmi = (1/k) · logₐb
Asosiy ayniyat	a^(logₐb) = b

Asosni almashtirish

• logₐb = lg b / lg a (asos 10 ga o'tkazish).
• logₐb va a ning b asosli logarifmi (b ≠ 1) o'zaro teskari: ko'paytmasi 1.
• logₐb · (c ning b asosli logarifmi) = logₐc.

1. Mavjudlik shartini tekshir (a > 0, a ≠ 1, b > 0)
2. Ifodani bir xil asosga keltir
3. Ko'paytma, bo'linma, daraja xossalarini qo'lla
4. Natijani hisobla

Misollar

• log₂32 + log₃9 = 5 + 2 = 7.
• log₆4 + log₆9 = log₆36 = 2.
• log₂3 · log₃8 = log₂8 = 3.
• lg 2 = a bo'lsa, lg 5 = lg(10/2) = 1 − a.
• 5^(log₅7 + 1) = 5^(log₅7) · 5 = 7 · 5 = 35.

Xatolar

• logₐ(b + c) ≠ logₐb + logₐc: yig'indining logarifmi uchun formula yo'q.
• logₐb · logₐc ≠ logₐ(bc): ko'paytma bilan yig'indini aralashtirmang.
• Manfiy yoki nol son logarifmi mavjud emas.