# Change of Base Logarithm formula in Reciprocal form

A mathematical rule of changing base of a logarithmic term by writing the term in its reciprocal form is called change of base logarithmic rule in reciprocal form.

## Formula

$\large \log_{b}{m} = \dfrac{1}{\log_{m}{b}}$

This change of base formula is used to express any logarithmic function in its reciprocal form.

### Proof

The change of base formula for logarithm in reciprocal form is derived in logarithmic mathematics by using the rules of exponents and mathematical relation between exponents and logarithms.

#### Basic step

$\log_{b}{m}$ and $\log_{d}{b}$ are two logarithmic terms and assume the values of them are $x$ and $y$ respectively.

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

Express the two logarithmic equations in exponential form according to mathematical relation between exponent and logarithm.

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

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

The change of base formula can be written in mathematical by this data.

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

#### Applying a technique

Assume $d = m$ and eliminate $d$ by $m$ in the change of base formula.

$\implies$ $\log_{b}{m} = \dfrac{\log_{m}{m}}{\log_{b}{m}}$

#### Obtaining the property

Mathematically, the logarithm of a number to same base is always one. Therefore, $\log_{m}{m} = 1$ and write the change of base formula.

$\,\,\, \therefore \,\,\,\,\,\,$ $\log_{b}{m} = \dfrac{1}{\log_{b}{m}}$