CAT 2025 Lesson : Logarithm - Concepts & Cheatsheet

Note: The video for this module contains a summary of all the concepts covered in this lesson. The video would serve as a good revision. Please watch this video in intervals of a few weeks so that you do not forget the concepts. Below is a cheatsheet that includes all the formulae but not necessarily the concepts covered in the video.

7. Cheatsheet

1

) If

n = a^{b}

, then

\mathrm{log}_{a} {n} = b

2

)

e^{x} = \displaystyle\sum_{n=0}^{\infty} \dfrac{x^{n}}{n!}

= \dfrac{x^{0}}{0!} + \dfrac{x^{1}}{1!} + \dfrac{x^{2}}{2!} + \dfrac{x^{3}}{3!} + ...

3

)

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	$a^{\mathrm{log} _{a}{x}} = x$

4

) Characteristic is the integer part, before the decimal point, of a logarithm of a number. Mantissa is the fractional part, after the decimal point, of a logarithm of a number.

5) If the characteristic of a common logarithm is a positive number, say

x

, then the number of digits of the number to the left of the decimal point is

x + 1

.

6) If the characteristic of a common logarithm is a negative number, say

x

, then the number of zeroes to the right of the decimal point and before the first non-zero digit is

x – 1

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