# Power Rule of Logarithm of an Exponential term to a number

## Formula

$\log_{\displaystyle b} m^{\displaystyle n} = n \log_{\displaystyle b} m$

Logarithm of a power of a number is equal to the logarithm of the number multiplied by the power.$### Proof$m$is a literal that represents the product of$x$literal factors of$b$and it is expressed as$b^{\displaystyle x} = m$in exponential form and it is also expressed as$x = \log_{\displaystyle b} m$in logarithmic system.$b^{\displaystyle x} = m \Leftrightarrow x = \log_{\displaystyle b} m$Take$n^{th}$power both sides of the equation$b^{\displaystyle x} = m$.$\implies {(b^{\displaystyle x})}^{\displaystyle n} = m^{\displaystyle n}$Use power rule of power of an exponential term to express it in simplified form.$\implies b^{\displaystyle xn} = m^{\displaystyle n}$Take$y = nx$and$z = m^{\displaystyle n}\implies b^{\displaystyle y} = z$Express this exponential form term in logarithmic form.$y = \log_{\displaystyle b} z$Replace the literals$y$and$z$by their respective values.$\implies nx = \log_{\displaystyle b} m^{\displaystyle n}\implies \log_{\displaystyle b} m^{\displaystyle n} = nx$We have already know that$b^{\displaystyle x} = m \Leftrightarrow x = \log_{\displaystyle b} m$. So, substitute the value of$x$.$\therefore \,\,\,\,\, \log_{\displaystyle b} m^{\displaystyle n} = n \log_{\displaystyle b} m$#### Verification Calculate the power a number.$\log_{e} 3^4 = \log_{e} 81 = 4.394449155 \cdots\implies \log_{e} 3^4 = 4.3944$Now, calculate the power times logarithm of the number.$\log_{e} 3 = 1.098612289 \cdots = 1.0986\implies 4 \times \log_{e} 3 = 4 \times 1.0986\implies 4 \times \log_{e} 3 = 4.3944\therefore \,\,\,\,\, \log_{e} 3^4 = 4 \times \log_{e} 3 = 4.3944\$

It is proved that logarithmic of power of a number is equal to the product of the exponent and logarithm of the number.