How do you simplify \[{e^{3\ln \left( x \right)}}\]?

Hint: Here, we will convert the given exponential function into the logarithmic function by using the conversion formula. We will then use logarithmic identities to simplify the given expression further and get the required answer. A logarithmic function is defined as the power to which a number must be raised to get the desired values.

Formula Used:
We will use the following formula:
1. Logarithmic Identity \[n\log m = \log {m^n}\].
2. Logarithmic Identity \[{m^{{{\log }_m}\left( n \right)}} = n\].

Complete step by step solution:
We are given a function \[{e^{3\ln \left( x \right)}}\].
Let \[f\left( x \right)\] be the given function.
\[ \Rightarrow f\left( x \right) = {e^{3\ln \left( x \right)}}\]
We know that the logarithmic identity \[n\log m = \log {m^n}\].
Now, we will simplify the expression by using the logarithmic identity, we get
\[ \Rightarrow f\left( x \right) = {e^{\ln \left( {{x^3}} \right)}}\]
We know that the logarithmic identity\[{m^{{{\log }_m}\left( n \right)}} = n\].
\[ \Rightarrow f\left( x \right) = {x^3}\]
Therefore, the expression after simplifying the given expression \[{e^{3\ln \left( x \right)}}\]is\[{x^3}\].

Additional Information:
If we are given that \[k = \ln \left( c \right)\] is the solution to the problem \[{e^k} = c\]. Thus
\[ \Rightarrow k = \ln \left( c \right)\]……………………………………………………………………………………….\[\left( 1 \right)\]
\[ \Rightarrow {e^k} = c\]……………………………………………………………………………………………..\[\left( 2 \right)\]
By substituting equation \[\left( 1 \right)\]in equation\[\left( 2 \right)\], we get

\[ \Rightarrow {e^{\ln \left( c \right)}} = c\]

Thus, the relationship between the natural logarithm with the base and the exponential function.

Note:
We know that a logarithmic equation is an equation that involves the logarithm of an expression with a variable on either of the sides. Logarithmic functions and exponential functions are inverses to each other. Common Logarithm is defined as the logarithm with the base 10. Common logarithm is denoted as \[{\log _{10}}x\]. Natural Logarithm is defined as the logarithm with the base \[e\]. Natural logarithm is denoted as \[\ln x\] or \[{\log _e}\left( x \right)\]. The natural logarithm of the logarithm with the base \[e\] is the inverse function of the natural exponential function.
The given expression is simplified by using the relationship between the natural logarithm with the base \[e\] and the exponential function. Thus, \[{e^{\ln \left( {{x^3}} \right)}} = {x^3}\].