Evaluate log of log of 10^100

Rewrite as an equation.

Rewrite ${\displaystyle \log \left(100\right)=x}$ in exponential form using the definition of a logarithm. If ${\displaystyle x}$ and ${\displaystyle b}$ are positive real numbers and ${\displaystyle b}$ does not equal ${\displaystyle 1}$, then ${\displaystyle {\log }_b\left(x\right)=y}$ is equivalent to ${\displaystyle b^y=x}$.

Create equivalent expressions in the equation that all have equal bases.

Since the bases are the same, the two expressions are only equal if the exponents are also equal.

The variable ${\displaystyle x}$ is equal to ${\displaystyle 2}$.