Ko‘rsatkichli, logarifmik va trigonometrik tenglamalar

Ko‘rsatkichli tenglamalar

Asosiy usul — ikkala tomonni bitta asosga keltirish: a^f(x) = a^g(x) ⇔ f(x) = g(x).

Misol. 4^x = 8 → 2^(2x) = 2³ → x = 3/2.

Yangi o‘zgaruvchi: 9^x − 4 · 3^x + 3 = 0, t = 3^x > 0: t² − 4t + 3 = 0 → t = 1 yoki t = 3 → x = 0 yoki x = 1.

Logarifmik tenglamalar

logₐ f(x) = c ⇔ f(x) = aᶜ, f(x) > 0 sharti bilan. Bir nechta logarifm bo‘lsa, avval hammasining aniqlanish sohasi yoziladi, keyin xossalar bilan bittaga yig‘iladi.

Misol. log₂ x + log₂(x − 2) = 3. Shart: x > 2. x(x − 2) = 8 → x² − 2x − 8 = 0 → x = 4 yoki x = −2. Shartga faqat x = 4 mos.

Trigonometrik tenglamalar

• sin x = a (|a| ≤ 1): x = (−1)ⁿ arcsin a + πn;
• cos x = a (|a| ≤ 1): x = ± arccos a + 2πn;
• tg x = a: x = arctg a + πn;
• xususiy hollar: sin x = 0 → x = πn; cos x = 0 → x = π/2 + πn; sin x = 1 → x = π/2 + 2πn.

Misol. 2cos²x − cos x − 1 = 0, t = cos x: t = 1 yoki t = −1/2. x = 2πn yoki x = ±2π/3 + 2πn.

|a| > 1 bo‘lsa sin x = a va cos x = a ning yechimi yo‘q.

Ko‘p uchraydigan xato

Logarifmik tenglamada aniqlanish sohasini tekshirmaslik va trigonometrik tenglamada yechimning davriy qismini (+2πn) tashlab ketish.