Proof of Power Law of Logarithms

Formula

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

Power Rule of logarithm reveals that log of a quantity in exponential form is equal to the product of exponent and logarithm of base of the exponential term.

$q$ is a quantity and it is expressed in exponential form as $m^{\displaystyle n}$. Therefore, $q \,=\, m^{\displaystyle n}$.

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

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

The value of logarithmic terms like $\log_{b}{(m^{\displaystyle n})}$ can be calculated by power law identity of logarithms. The power law property is actually derived by the power rule of exponents and relation between exponent and logarithmic operations.

Express quantity in Exponential form

Take, $m$ is a quantity and it is expressed in exponential form on the basis of another quantity $b$. The total number of multiplying factors of $b$ is $x$ for representing the quantity $m$.

$m$ $\,=\,$ $\underbrace{b \times b \times b \times \ldots \times b}_{\displaystyle x \, factors}$

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

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

Take Power Both sides

$n$ is a quantity and take $n$-th power both sides of the exponential form equation.

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

As per power rule of exponents, the whole power of a quantity in exponential form is equal to base is raised to the power of product of exponents.

$\implies b^{\displaystyle nx} \,=\, m^{\displaystyle n}$

Express exponential form quantity in logarithm form

Take $y \,=\, nx$ and $z \,=\, m^{\displaystyle n}$.

$\implies b^{\displaystyle y} \,=\, z$

Express equation in logarithmic form as per the mathematical relation between exponents and logarithms.

$\implies y \,=\, \log_{b}{z}$

Replace the literals $y$ and $z$ by their respective values.

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

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

Property of Power Law in Algebraic form

It is taken in the first step that $b^{\displaystyle x} \,=\, m$ and it can be expressed in logarithmic form as $x \,=\, \log_{b}{m}$.

So, replace the value of $x$ in logarithmic form in the above equation for deriving the power rule of logarithms in algebraic form.

$\,\,\, \therefore \,\,\,\,\,\, \log_{b}{(m^{\displaystyle n})}$ $\,=\,$ $n\log_{b}{m}$

Therefore, it is proved that logarithm of a quantity in exponential form is equal to product of the exponent and log of the base of the exponent.

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