Power Law of Logarithm of a number to an Exponential term

Formula

$\log_{\displaystyle b^n} m = \dfrac{1}{n} \log_{\displaystyle b} m$

Proof

$m$ and $b$ are two literal numbers. The logarithm of $m$ to the base $b$ is written as $\log_{\displaystyle b} m$.

If the base of the logarithm is raised to the power of another number $n$, then it is expressed as $b^n$ in exponential form. If the value of the logarithm of the literal $m$ is required to calculate for the base $b$ raised to the power $n$, then it is written in logarithmic form as $\log_{\displaystyle b^n} m$

A logarithmic formula is essential to express $\log_{\displaystyle b^n} m$ in terms of $\log_{\displaystyle b} m$.

Take $\log_{\displaystyle b^n} m$ is equal to $y$.

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

Express this logarithmic expression in exponential notation.

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

Use power law of exponents to express the exponent of an exponential term as an exponential term.

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

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

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

Write this exponential expression in logarithmic form.

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

Now, replace the $x$ by its actual value.

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

Similarly, replace the value of $y$ in this logarithmic expression.

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

$\therefore \,\,\,\,\, \log_{\displaystyle b^n} m = \dfrac{1}{n} \log_{\displaystyle b} m$

The property of the logarithm is used to get the value of the logarithm of a number to a base which can be expressed as an exponent of a number by dividing the value of logarithm of a number to a base by the exponent of the base.

Verification

For example, $\log_{125} 2 = 0.14355\cdots \approx 0.14356$

The value can also be obtained in alternative method.

$\log_{125} 2 = \log_{\displaystyle 5^3} 2$

$\implies \log_{\displaystyle 5^3} 2 = \dfrac{1}{3} \log_{5} 2$

$\implies \log_{\displaystyle 5^3} 2 = \dfrac{1}{3} \times 0.43068$

$\implies \log_{\displaystyle 5^3} 2 = \dfrac{0.43068}{3}$

$\implies \log_{\displaystyle 5^3} 2 = 0.14356$

Compare the both values to understand this fundamental law of logarithm in mathematics.

$\therefore \,\,\,\,\,\, \log_{125} 2$ $=$ $\log_{\displaystyle 5^3} 2$ $=$ $\dfrac{1}{3} \log_{\displaystyle 5} 2 = 0.14356$


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