Math Doubts

Change of Base Rule of Logarithms


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

The base of a logarithmic term can be changed mathematically by expressing it as a quotient of two logarithmic terms which contain another quantity as their base. It is usually called as change of base rule and used as a formula in logarithms.


The change of base log formula in quotient form is derived in algebraic form on the basis of rules of exponents and also mathematical relation between exponents and logarithms.

Basic step

$\log_{b}{m}$ and $\log_{d}{b}$ are two logarithmic terms and it is taken that their values are $x$ and $y$ respectively.

$\log_{b}{m} = x$ and $\log_{d}{b} = y$

Express both logarithmic equations in exponential form by the mathematical relationship between them.

$(1) \,\,\,$ $\log_{b}{m} = x \,\Leftrightarrow\, m = b^{\displaystyle x}$

$(2) \,\,\,$ $\log_{d}{b} \,\,\, = y \,\Leftrightarrow\, b = d^{\displaystyle y}$

Changing the Base of Exponential term

Eliminate the base $b$ in the equation $m = b^{\displaystyle x}$ by substituting the $b = d^{\displaystyle y}$.

$\implies$ $m = {(d^{\displaystyle y})}^{\displaystyle x}$

Apply power of power exponents rule to simplify this exponential equation.

$\implies$ $m = d^{\displaystyle xy}$

Express this exponential equation in logarithmic form.

$m = d^{\displaystyle xy} \Longleftrightarrow xy = \log_{d}{m}$

Obtaining the property

$\implies$ $xy = \log_{d}{m}$

In fact, $x = \log_{b}{m}$ and $y = \log_{d}{b}$. So, replace them.

$\implies$ $\log_{b}{m} \times \log_{d}{b} = \log_{d}{m}$

$\,\,\, \therefore \,\,\,\,\,\,$ $\log_{b}{m} = \dfrac{\log_{d}{m}}{\log_{d}{b}}$

Thus, the change of base formula is derived in mathematics for changing base of any log term.

Math Questions

The math problems with solutions to learn how to solve a problem.

Learn solutions

Math Worksheets

The math worksheets with answers for your practice with examples.

Practice now

Math Videos

The math videos tutorials with visual graphics to learn every concept.

Watch now

Subscribe us

Get the latest math updates from the Math Doubts by subscribing us.

Learn more

Math Doubts

A free math education service for students to learn every math concept easily, for teachers to teach mathematics understandably and for mathematicians to share their maths researching projects.

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved