How do you solve \[\ln \left( {{x}^{2}} \right)=16\]?

Hint: The natural logarithm (ln) is the logarithm to the base of e and also called the inverse of the exponential function. ‘e’ is an irrational number that is a constant and its value is 2.718281828459. It can also be written as \[{{\log }_{e}}x\]. \[\ln x\]is undefined when \[x\le 0\].

Complete step by step answer:
As per the given question, we need to find the value of x by solving the expression \[\Rightarrow \]\[\ln \left( {{x}^{2}} \right)=16\].
According to the properties of the logarithm
\[\Rightarrow \ln {{x}^{a}}=a\ln x\]
From this property the left hand side of the expression will be
\[\Rightarrow \ln \left( {{x}^{2}} \right)=2\ln x\]
we can rewrite the above expression as
\[\Rightarrow 2\ln x=16\]
Now we divide the above expression with 2 on both sides then the equation becomes
\[\Rightarrow \dfrac{2}{2}\ln x=\dfrac{16}{2}\]
We know that 16 is a multiple of 2. Then the expression will be
\[\Rightarrow \ln x=8\]
Now we apply exponential function. Then the expression becomes
\[\Rightarrow \]\[{{e}^{\ln x}}={{e}^{8}}\]
Since \[\ln x\] has base e from properties of logarithm \[{{a}^{{{\ln }_{a}}x}}=x\].
\[\Rightarrow x={{e}^{8}}\]
The value of \[{{e}^{8}}\] is \[2980.95798704\].
We can round off the value to 3 decimal places. Then the value will be
\[\Rightarrow x={{e}^{8}}=2980.958\]
Therefore, in this way we can solve the given expression \[\ln \left( {{x}^{2}} \right)=16\].
Therefore, the value of x is \[2980.958\].

Note:
In order to solve these types of problems, we need to have knowledge of logarithms and their properties. The value of an exponential function can be known by using a calculator. We should know all the properties of logarithms and exponentials so that we can solve the expression easily. We should avoid calculation mistakes to get the correct solution.