Logarifmlar

🟢 — masalani tezlashtiradi · 🟡 — ilg'or daraja · 🔵 — chuqur o'rganish (olimpiada darajasi)

🟢 Ko'rsatkichli tenglamalar (bir xil asosga keltirish)

$a > 0$, $a \ne 1$ bo'lsa: $\;a^x = a^y \;\Rightarrow\; x = y$

Misol: $4^x - 3 \cdot 2^x + 2 = 0$. $u = 2^x$ desak, $u^2 - 3u + 2 = 0$, $u = 1$ yoki $u = 2$, ya'ni $x = 0$ yoki $x = 1$.

🟡 Logarifm qoidalari

$\log_a(xy) = \log_a x + \log_a y \qquad \log_a\frac{x}{y} = \log_a x - \log_a y \qquad \log_a x^n = n\log_a x$

$\log_a a = 1 \qquad \log_a 1 = 0 \qquad a^{\log_a x} = x \qquad \log_a b = \frac{\ln b}{\ln a}$

• $a^x = b \;\Rightarrow\; x = \dfrac{\ln b}{\ln a}$

🟡 Ko'rsatkichli va logarifmik juftlik

$y = e^x \;\leftrightarrow\; y = \ln x \qquad e^{\ln x} = x \;(x > 0) \qquad \ln e^x = x$