# Switch Rule of Logarithms

## Formula

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

The quantity and base in a logarithmic term can be switched by changing the base in reciprocal form. It is called as base switch rule of logarithms and it is used as a formula in logarithmic mathematics.

### 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 it is assumed that the values of them are $x$ and $y$ respectively.

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

The equations in logarithmic form can be written in exponential form by the mathematical relationship between exponents and logarithms.

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

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

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

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

#### Applying a technique

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

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

#### Obtaining the property

Mathematically, the logarithm of a quantity (equals to base) is always one as per log base rule. Therefore, $\log_{m}{m} = 1$ and write the change of base formula.

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

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Jun 26, 2023

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