Logarithm of an exponential term

The system of logarithm is introduced in mathematics based on principle of the exponentiation. Exponential form terms are appeared in logarithmic system. So, a logarithmic identity is essential to simply such expressions. Actually, logarithm of an exponential term is equal to product of exponent and logarithmic of the base and it is proved here in algebraic form.

Proof

Assume, $b,x$ and $m$ are three algebraic factors. Assume, the value of $b$ raised to the power $x$ is equal to $m$ and it is expressed in algebraic form. According to logarithmic system, the relation between an exponential term ${b}^{x}$ and $m$ can also be written in logarithmic form.

${b}^{x}=m⇔x={log}_{b}m$

Assume, $n$ is another algebraic factor and it has a relation with algebraic factor $m$ in exponential form. In other words, ${m}^{n}$.

${m}^{n}$

Actually, the value of $m$ is equal to ${b}^{x}$. So, replace $m$ by its original value.

${m}^{n}={\left({b}^{x}\right)}^{n}$

According to power rule of exponentiation, exponent of an exponential term is equal to product of exponents of same base.

$⇒{m}^{n}={b}^{x×n}$

Assume, $x×n=y$ and ${m}^{n}=z$

$⇒{m}^{n}={b}^{x×n}$

$⇒z={b}^{y}$

Apply principle of logarithm to this exponential term.

${b}^{y}=z⇔y={log}_{b}z$

Now, replace the algebraic factors $y$ and $z$ by their actual values.

$y={log}_{b}z$

$⇒x×n={log}_{b}{m}^{n}$

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

$⇒\left({log}_{b}m\right)×n={log}_{b}{m}^{n}$

It can be written as follows.

$⇒{log}_{b}{m}^{n}=n×{log}_{b}m$

It is proved that logarithm of an exponential term is equal to product of exponent and logarithm of same base.

Example

${3}^{4}$

It is an exponential form term. Logarithm of exponential term can be found by using the logarithmic system.

${log}_{e}{3}^{4}={log}_{e}81$

$⇒{log}_{e}{3}^{4}=4.394449155$

Now, calculate the $4$ times of the logarithm of $3$.

$4×{log}_{e}3$ $=$ $4×1.098612289$

$⇒4×{log}_{e}3$ $=4.394449155$

Therefore,

${log}_{e}{3}^{4}=4×{log}_{e}3$ $=4.394449155$

The value of logarithm of an exponential term is equal to product of the exponent and logarithm of the same base.

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