Math Doubts

Proof of Base Power Law of Logarithms


$\log_{b^n}{m}$ $\,=\,$ $\Big(\dfrac{1}{n}\Big)\log_{b}{m}$

The base power rule of logarithm expresses that the value of log of a quantity when its base is in exponential form, is equal to the quotient of logarithm of quantity to base of exponential term by the exponent.

$q$ is a quantity and it is written in exponential notation as $b^{\displaystyle n}$. Therefore, $q \,=\, b^{\displaystyle n}$.

The log of a quantity to a base ($q$) is written as $\log_{q}{m}$. The value of $\log_{q}{m}$ can be calculated by calculating $\log_{b^n}{m}$.

$\log_{q}{m}$ $\,=\,$ $\log_{b^n}{m}$

The values of logarithmic terms like $\log_{b^n}{m}$ can be calculated by using base power law identity of logarithms. The property of power law is originally derived by the power rule of exponents and the relation between logarithmic and exponents operations.

Value of Logarithm of a Quantity

$m$ is a quantity and it is divided as multiplying factors of another quantity whose value is expressed in exponential form as $b^{\displaystyle n}$. Therefore, the number of multiplying factors is represented by $\log_{b^n}{m}$ in mathematics.

Take the value of $\log_{b^n}{m}$ is equal to $y$.

$\log_{b^n}{m} \,=\, y$

Express Logarithmic relation in exponential form

The logarithmic equation can be transformed into exponential form equation by the relation between exponents and logarithms.

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

According to Power Rule of exponents, the value of whole power of an exponential term is equal to base is raised to the power of product of the exponents.

$\implies m \,=\, b^{\displaystyle ny}$

Write Exponential form relation in Logarithms

Take $x = ny$ and replace exponent $ny$ by $x$ in the expression.

$\implies m \,=\, b^{\displaystyle x}$

Write this exponential form equation in logarithmic form.

$\implies x \,=\, \log_{b}{m}$

Now, replace the actual value of $x$ in the logarithmic equation.

$\implies ny \,=\, \log_{b}{m}$

Get the Property of Base Power Rule of Logarithms

It is taken in the first step of the derivation that $\log_{b^n}{m} \,=\, y$. So, replace the value of $y$ by its actual value in the logarithmic equation.

$\implies n\log_{b^n}{m} \,=\, \log_{b}{m}$

$\,\,\, \therefore \,\,\,\,\,\, \log_{b^n}{m} \,=\, \Big(\dfrac{1}{n}\Big)\log_{b}{m}$

It is a power rule of logarithms but it is used when base of a logarithmic term is in exponential form. Hence, it is also called as base power rule of logarithms.

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