Math Doubts

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.

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

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



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