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Change of Base Logarithm formula in Product form

A rule of changing base of a logarithmic term by expressing it as a product of two logarithmic terms is called the change of base logarithmic rule in product form.

Formula

$\large \log_{b} m = \log_{a} m \times \log_{b} a$

It is used as a formula to split a logarithmic term as a product of two logarithmic terms in which one term has different base and second term has the number which is the base of the first term. It is also used in inverse mathematical operation.

Proof

The change of base log formula in product form can be derived in algebraic form by considering rules of exponents and the mathematical relation between logarithms and exponents.

Basic step

$\large \log_{b}{m}$, $\large \log_{a}{m}$ and $\large \log_{b}{a}$ are three logarithmic terms and their values are $x$, $y$ and $z$ respectively. Express three of them in exponential form as per the mathematical relation between exponents and logarithms.

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

$(2) \,\,\,$ $\log_{a}{m} = y \,\Longleftrightarrow\, m = a^{\displaystyle y}$

$(3) \,\,\,$ $\log_{b}{a} \,\,\, = z \,\Longleftrightarrow\, a = b^{\displaystyle z}$

Changing the Base

$m = a^{\displaystyle y}$. Change the base of exponential term of this equation on the basis of exponential equation $a = b^{\displaystyle z}$.

$\implies$ $m = {(b^{\displaystyle z})}^{\displaystyle y}$

Use power of power exponents rule to simplify it.

$\implies$ $m = b^{\displaystyle yz}$

Transform this exponential equation in logarithmic form.

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

Obtaining the property

Actually, $\log_{a}{m} = y$ and $\log_{b}{a} = z$. Therefore, replace them to obtain the change of base log formula.

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

$\,\,\, \therefore \,\,\,\,\,\,$ $\log_{b}{m} = \log_{a}{m} \times \log_{b}{a}$

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