Ko‘rsatkichli ifodalar va logarifm

Ta’rif

logₐ b = c ⇔ aᶜ = b, bunda a > 0, a ≠ 1, b > 0. Logarifm — «a ni qanday darajaga ko‘tarsak b chiqadi?» degan savolga javob. log₂ 8 = 3, chunki 2³ = 8.

Asosiy ayniyat: a^(logₐ b) = b. lg — o‘nli (asos 10), ln — natural logarifm (asos e ≈ 2,718).

Xossalar

• logₐ(xy) = logₐ x + logₐ y;
• logₐ(x/y) = logₐ x − logₐ y;
• logₐ(xᵖ) = p · logₐ x;   log_(aᵏ) x = (1/k) logₐ x;
• asosni almashtirish: logₐ b = log_c b / log_c a;
• logₐ b · log_b a = 1;   logₐ 1 = 0;   logₐ a = 1.

Misol. log₆ 4 + log₆ 9 = log₆ 36 = 2.

Misol. log₂ 5 · log₅ 32 = log₂ 32 = 5 (asosni almashtirish zanjiri).

Ko‘rsatkichli ifodalar

Daraja xossalari logarifm bilan birga ishlatiladi: 3^(2 + log₃ 5) = 3² · 3^(log₃ 5) = 9 · 5 = 45.

Ko‘rsatkichli funksiya y = aˣ: a > 1 da o‘sadi, 0 < a < 1 da kamayadi; qiymatlari doim musbat. Logarifmik funksiya y = logₐ x unga teskari, aniqlanish sohasi x > 0.

Ko‘p uchraydigan xato

• logₐ(x + y) = logₐ x + logₐ y deb yozish — xossa faqat ko‘paytmaga tegishli.
• Logarifm ostidagi ifodaning musbatligini tekshirmaslik.