Change of Base Rule of Logarithms

Formula

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

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.

Proof

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.