How does $e\left( {2.718} \right)$ help in applications or implications in real-life.

Hint: $e\left( {2.718} \right)$ is an irrational number and is the base of natural logarithmic number, also it’s the natural exponential function. So in order to find the applications of $e\left( {2.718} \right)$ we have to find the different fields in which it is used and the importance of it in those fields.

Complete step by step solution:
Now there are few applications and implications of $e\left( {2.718} \right)$ real life as discussed below:
In Population Models:
Consider a population with$p$people and that this population doubles every 30 years. After 180 years, say, the population will double:
\[\dfrac{{180}}{{30}} = 6\,{\text{times}}\].
Now let the population after 180 years be $P$. Then $P$:
$P = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times p = {2^6}p.................\left( i \right)$
Now to find the instantaneous rate of growth of the population to compare it to former rates and know what it would be in future.
Here is when we can use $e$.
Now population after $t$ years is going to be: $P = {2^{\left( {\dfrac{t}{{30}}} \right)}}p$
Now after an infinitesimal amount of time the growth of population is represented by the instantaneous rate of change.
So let the infinitesimal amount of time be $dt$ and the effect of this in the population be $dP$. Then we can write:
$\dfrac{{dP}}{{dt}} = \dfrac{{p \times {{\log }_e}2}}{{30}} \times {2^{\dfrac{t}{{30}}}} = \dfrac{{p
\times \ln 2}}{{30}} \times {2^{\dfrac{t}{{30}}}}...............\left( {ii} \right)$
Here $d$ is not a constant but it represents the infinitesimal quantity.
Apart from this $e\left( {2.718} \right)$ appears in decay and growth conditions.
In Physics: Boltzmann distribution: In statistical mechanics it’s used a probability measure that gives the probability that a system will be in a certain state in terms of that state’s energy and the temperature of the system.
Such that for a system having 2 states the probability ${p_1}\,\,{\text{and}}\,\,{p_2}$ can be written as:
\[
{p_1} = \dfrac{{{e^{ - \dfrac{{{E_1}}}{{kT}}}}}}{{{e^{ - \dfrac{{{E_1}}}{{kT}}}} + {e^{ -
\dfrac{{{E_2}}}{{kT}}}}}}....................\left( {iii} \right) \\
{p_1} = \dfrac{{{e^{ - \dfrac{{{E_2}}}{{kT}}}}}}{{{e^{ - \dfrac{{{E_1}}}{{kT}}}} + {e^{ -
\dfrac{{{E_2}}}{{kT}}}}}}....................\left( {iv} \right) \\
\]
$
T = {\text{Temperature}} \\
{{\text{E}}_1}\,{\text{and}}\,{{\text{E}}_2} = {\text{Different}}\,{\text{Energies}} \\
k\,\,\,\,\,\,{\text{ = Boltzmann constant}} \\
$
Note: Apart from all the above mentioned ones $e\left( {2.718} \right)$ is extensively used in calculating growth and decay problems. Due to $e\left( {2.718} \right)$ we have witnessed many notable breakthroughs in the scientific field. Also $e\left( {2.718} \right)$ is extensively used in the field of finance which is an unexpected field but it plays a major role in that field.