Laws of Logarithms

Log One Rule

Logarithm of one to any number is zero.

$\large \log_{b}{1} = 0$

Log of base Rule

Logarithm of base quantity is always one.

$\large \log_{b}{b} \,=\, 1$

Fundamental Law

$\large b^\log_{b} m} =$

Product Law

Logarithm of the product of two or more quantities is equal to sum of their logs.

$\large (1) \,\,\,\,\,\,$ $\large \log_{b}{(m.n)}$ $\large \,=\,$ $\large \log_{b} m + \log_{b} n$

$\large (2) \,\,\,\,\,\,$ $\large \log_{b}{(m.n.o \cdots)}$ $\large \,=\,$ $\large \log_{b}{m}$ $+$ $\large \log_{b}{n}$ $+$ $\large \log_{b}{o} + \cdots$

Quotient Rule

Logarithm of quotient of two numbers is equal to difference of their logs.

$\large \log_{b}{\Big(\dfrac{m}{n}\Big)}$ $\large \,=\,$ $\large \log_{b}{m} \,–\, \log_{b}{n}$

Power Rules

The rules of logarithms which simplify the way to find the values of logarithmic terms by expressing quantity and base quantity of log terms in exponential form.

$\large (1). \,\,\,\,\,\,$ $\large \log_{b}{m^n} \,=\, n\log_{b}{m}$

$\large (2). \,\,\,\,\,\,$ $\large \log_{b^y}{m} \,=\, {\Big(\dfrac{1}{y}\Big)}\log_{b}{m}$

$\large (3). \,\,\,\,\,\,$ $\large \log_{b^y}{m^x} \,=\, {\Big(\dfrac{x}{y}\Big)}\log_{b}{m}$

07

Change of Base formulas

$\large (1). \,\,\,\,\,\,$ $\large \log_{b}{m} = \log_{d}{m} \times \log_{b}{d}$

$\large (2). \,\,\,\,\,\,$ $\large \log_{b}{m} = \dfrac{\log_{d}{m}}{\log_{d}{b}}$

$\large (3). \,\,\,\,\,\,$ $\large \log_{b}{m} = \dfrac{1}{\log_{m}{b}}$

Email subscription
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
Follow us on Social Media
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.