How do you simplify $\ln \left( {\ln {e^{{e^{10}}}}} \right)$?

Hint: To solve this problem we will use the fact that $\ln \left( x \right)$ and ${e^x}$ are inverse of each other. That is, in simpler words we can say that $\ln \left( {{e^x}} \right) = x$. Now, we will use this property to solve the given problem. In the question, there are two such types of function, so we will have to use this property two times in order to solve the problem. So, let us see how to solve this problem.

Complete step by step answer:
Now, the given function is, $\ln \left( {\ln {e^{{e^{10}}}}} \right)$.
Let us assume that $\ln {e^{{e^{10}}}} = t$.
Using the property $\ln \left( {{e^x}} \right) = x$ on $t$, we get,
 $t = \ln {e^{{e^{10}}}}$
$ \Rightarrow t = {e^{10}}$
Therefore, we can write the original function as,
$\ln \left( {\ln {e^{{e^{10}}}}} \right) = \ln \left( t \right) = \ln \left( {{e^{10}}} \right)$.
Now, again using the property $\ln \left( {{e^x}} \right) = x$, we get,
$\ln \left( {\ln {e^{{e^{10}}}}} \right) = \ln \left( {{e^{10}}} \right) = 10$.
Therefore, the value of $\ln \left( {\ln {e^{{e^{10}}}}} \right)$ is $10$.

Note: Logarithmic functions are very important in order to convert equations into simpler and easier forms to solve and operate. These are directly related to the exponential function as inverse, as mentioned above. Logarithmic functions are helpful in every sector of mathematics, may it be calculus, algebra, real analysis, etc. For instance, in calculus, complex differentiations can be converted into simpler and easier to solve forms by taking logarithmic functions on both sides. Logarithmic functions gives various values based on the base of the $\log $. Like, ${\log _{10}}10 = 1$, where the base of $\log $ is $10$, while ${\log _e}e = 1$ or $\ln e = 1$, where $e$ is the base of $\log $. (Logarithmic function with base $e$ is called natural log and written as $\ln $).