Math Doubts

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.

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



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