CAT 2025 Lesson : Logarithm - Log Properties

3. Logarithm Properties

S.No.	Rule
1	$\mathrm{log} _{a}{x}$ is defined where (i) $x$ and $a$ are real numbers, (ii) $x > 0$ , (iii) $a > 0$ and $a \ne 1$
2	$\mathrm{log} _{a}{1} = 0$
3	$\mathrm{log} _{a}{a} = 1$
4	$\mathrm{log} \ a + \mathrm{log} \ b = \mathrm{log} \ ab$
5	$\mathrm{log} \ a$ $-\ \mathrm{log} \ b$ $= \mathrm{log} \ \left( \dfrac{a}{b}\right)$
6	$\mathrm{log} \ a^{b} = b \ \mathrm{log} \ a$
7	$\mathrm{log} _{b}{a} = \dfrac{1}{\mathrm{log} _{a}{b}}$
8	$\mathrm{log} _{b^{n}}{a^{m}} = \dfrac{m}{n} \times \mathrm{log} _{b}{a}$
9	$\mathrm{log} _{b}{a} = \mathrm{log} _{c}{a} \times \mathrm{log} _{b}{c}$
10	$\mathrm{log} _{b}{a}$ $= \dfrac{\mathrm{log} _{c}{a}}{\mathrm{log} _{c}{b}}$
11	$a^{\mathrm{log} _{a}{x}} = x$

Brief explanations for some of the above rules are given below.

Rule 2: Let $\mathrm{log} _{a} {1} = x$ ⇒ $1 = a^{x}.$ Therefore, $\bm{x = 0}$

Rule 3: Let $\mathrm{log} _{a} {a}$ ⇒ $a = a^{x}$. Therefore, $\bm{x = 1}$.

Rule 6: $\mathrm{log} \ a^{b}$ $= \mathrm{log} \ {(a \times a \times ..._{b times})}$ $= \mathrm{log} \ {a} + \mathrm{log} \ {a} + ... _{b times}$ $=b \ \mathrm{log} \ a$

Rule 8: $\mathrm{log} _{b^{n}} {a^{m}} = m\mathrm{log} _{b^n}{a}$ $= \dfrac{m}{\mathrm{log} _{a}{b^{n}}}$ $= \dfrac{m}{n\mathrm{log} _{a} {b}}$ $= \dfrac{m}{n} \times \mathrm{log} _{b}{a}$

Rule 11: Let $a^{\mathrm{log} _{a}{x}} = y$
Taking log to the base $a$ on both sides,
$\mathrm{log} _{a}{x} \times \mathrm{log} _{a}{a}$ $= \mathrm{log} _{a}{y}$
⇒ $\mathrm{log} _{a}{x} = \mathrm{log} _{a}{y}$ ⇒ $x = y$
⇒ $x = a^{\mathrm{log} _{a}{x}}$