# Change of Base Logarithm formula in Quotient form

A mathematical rule of changing base of a logarithmic term by expressing it as a quotient of two logarithmic terms is called change of base logarithmic rule in quotient form.

## Formula

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

It is used as a formula to divide any logarithmic term as a quotient of two logarithmic terms which contain same base and this change of base formula is also used in reverse operation.

### 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 their values are $x$ and $y$ respectively.

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

Express both logarithmic equations in exponential form as per the mathematical relation between exponent and logarithm.

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

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

#### 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.

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

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