# 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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