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Parin Sir > Quantitative Aptitude (Maths) > Logarithm > Logarithm

Logarithm

June 6, 2020
Category: Logarithm

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Introduction

☼ Logarithms : If x is a positive real number, other than 1 and $x^{y}$ = z , then y is called the logarithm of z to the base x, written as $\textup{log}_{x}\textup{z}$ .

Thus, $x^{y}$ = z $\Leftrightarrow$ y = $\textup{log}_{x}\textup{z}$ ; Where, x $\in$ $\textup{R}^{+}$ – { 1 } & z $\in$ $\textup{R}^{+}$

$\mapsto$ log x = $\textup{log}_{10}\textup{ x}$ ( logarithm to the base 10 is called common logarithm. )

( i.e. If the base of logarithm is not mentioned then it is taken as 10. )

$\mapsto$ In Higher Studies, ln x = $\log _{e} x$ & log x = $\log _{10} x$

$\mapsto$ In calculus, log x = $\log _{e} x$

( i.e. If the base of logarithm is not mentioned then it is taken as e. )

e.g. (i) $10^{3}$ = 1000 $\large \Leftrightarrow$ 3 = $\textup{log}_{10}\textup{ }1000$

(ii) $5^{2}$ = 25 $\large \Leftrightarrow$ 2 = $\textup{log}_{5}\textup{ }25$

(iii) $2^{-3}$ = $\frac{1}{8}$ $\large \Leftrightarrow$ – 3 = $\textup{log}_{2}$ $\frac{1}{8}$

(iv) $\left (0.1 \right )^{5}$ = 0.00001 $\large \Leftrightarrow$ 5 = $\textup{log}_{0.1}$ 0.00001

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