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.


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