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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