Proof of Fundamental Power Logarithmic identity

The logarithm of an exponential form quantity is equal to the product of the exponent and the logarithm of base of exponential quantity as per the fundamental power law of the logarithms.

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

Let’s learn how to prove the power rule of logarithms fundamentally in algebraic form.

Express a quantity in exponential notation

Let $m$ represents a quantity and it is split as the factors on the basis of a literal quantity $b$. The mathematical relationship between them can be written as follows.

$m$ $\,=\,$ $b \times b \times b \times \ldots \times b$

Let us assume that the total number of factors is denoted by a literal quantity $x$.

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

According to the exponentiation, the product of the factors can be written in exponential notation as follows.

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

Now, let’s assume that the literal quantity $m$ is raised to the power of $n$. Therefore, the exponential quantity $b$ raised to the power of $x$ should also be raised to the power of $n$.

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

Use the power rule of the exponents to find the power of the exponential quantity.

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

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

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

Write exponential notation in Logarithmic form

Let us represent $y \,=\, nx$ and $z \,=\, m^{\displaystyle n}$ for our convenience. Now, write the mathematical equation in terms of $y$ and $z$.

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

Now, use the math relationship between the exponents and logarithms to write the above equation in logarithmic form.

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

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

Replace variables with actual values in equation

The logarithmic equation is obtained in terms of the variables $y$ and $z$ but they are assumed values. So, replace them by their corresponding values in the equation.

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

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

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

It is time to express the value of variable $x$ in terms of literal quantities. We have taken that the value of $m$ is equal to the $b$ raised to the power of $x$ and it is written in the following mathematical form.

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

Now, use the mathematical relation between the exponents and logarithms to write the above exponential equation as a logarithmic equation.

$m \,=\, b^{\displaystyle x}$ $\Longleftrightarrow$ $x \,=\, \log_{b}{m}$

It is evaluated that the value of $x$ is equal to the logarithm of $m$ to the base $b$. Now, replace the value of $x$ in the logarithmic equation.

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

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

Therefore, it is derived that the logarithm of $m$ raised to the power of $n$ to the base $b$ is equal to the $n$ times logarithm of $m$ to base $b$. This mathematical property is called the fundamental power logarithmic identity in algebraic form.