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CAT 2025 Lesson : Logarithm - Concepts & Cheatsheet

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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

11) If n=abn = a^{b}, then logan=b\mathrm{log}_{a} {n} = b

22) ex=n=0xnn!e^{x} = \displaystyle\sum_{n=0}^{\infty} \dfrac{x^{n}}{n!} =x00!+x11!+x22!+x33!+...= \dfrac{x^{0}}{0!} + \dfrac{x^{1}}{1!} + \dfrac{x^{2}}{2!} + \dfrac{x^{3}}{3!} + ...

33)
S.No. Rule
1 logax\mathrm{log} _{a}{x} is defined where (i) xx and aa are real numbers, (ii) x>0x > 0 , (iii) a>0a > 0 and a1a \ne 1
2 loga1=0\mathrm{log} _{a}{1} = 0
3 logaa=1\mathrm{log} _{a}{a} = 1
4 log a+log b=log ab\mathrm{log} \ a + \mathrm{log} \ b = \mathrm{log} \ ab
5 log a\mathrm{log} \ a  log b-\ \mathrm{log} \ b =log (ab)= \mathrm{log} \ \left( \dfrac{a}{b}\right)
6 log ab=b log a\mathrm{log} \ a^{b} = b \ \mathrm{log} \ a
7 logba=1logab\mathrm{log} _{b}{a} = \dfrac{1}{\mathrm{log} _{a}{b}}
8 logbnam=mn×logba\mathrm{log} _{b^{n}}{a^{m}} = \dfrac{m}{n} \times \mathrm{log} _{b}{a}
9 logba=logca×logbc\mathrm{log} _{b}{a} = \mathrm{log} _{c}{a} \times \mathrm{log} _{b}{c}
10 logba\mathrm{log} _{b}{a} =logcalogcb= \dfrac{\mathrm{log} _{c}{a}}{\mathrm{log} _{c}{b}}
11 alogax=xa^{\mathrm{log} _{a}{x}} = x


44) 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
xx, then the number of digits of the number to the left of the decimal point is x+1x + 1.

6) If the characteristic of a common logarithm is a negative number, say
xx, then the number of zeroes to the right of the decimal point and before the first non-zero digit is x1x – 1.

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