rag/knowledge_base/logarithms_exponents.md

Logarithms and Exponents

Exponent Laws

• $a^m \cdot a^n = a^{m+n}$
• $\frac{a^m}{a^n} = a^{m-n}$
• $(a^m)^n = a^{mn}$
• $(ab)^n = a^n b^n$
• $a^0 = 1$ for $a \neq 0$
• $a^{-n} = \frac{1}{a^n}$
• $a^{1/n} = \sqrt[n]{a}$

Logarithm Definition

$\log_a x = y \iff a^y = x$ where $a > 0, a \neq 1, x > 0$

Logarithm Properties

• $\log_a(xy) = \log_a x + \log_a y$
• $\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$
• $\log_a(x^n) = n \log_a x$
• $\log_a a = 1$
• $\log_a 1 = 0$

Change of Base

$\log_a x = \frac{\log_b x}{\log_b a} = \frac{\ln x}{\ln a}$ $\log_a b = \frac{1}{\log_b a}$ $\log_a b \cdot \log_b c = \log_a c$

Common Logarithm Identities

• $a^{\log_a x} = x$
• $\log_{a^k} x = \frac{1}{k} \log_a x$
• $\log_a x = \log_a y \implies x = y$

JEE Tips

• Domain of $\log_a f(x)$: need $f(x) > 0$
• $\log_a x > \log_a y$: if $a > 1$ then $x > y$; if $0 < a < 1$ then $x < y$
• When solving $\log$ equations, always verify solutions satisfy domain constraints