CAT 2025 Lesson : Logarithm - Log 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}}

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