Sure, let's work through this step-by-step using the Laws of Logarithms.
1. Evaluate the inner expression [tex]\(10^{1000}\)[/tex]:
[tex]\[ 10^{1000} = \underbrace{1000~\text{zeros}}_{\text{1 followed by}} \][/tex]
2. Calculate the logarithm of the inner expression:
[tex]\[ \log_{10}(10^{1000}) \][/tex]
Using the power rule of logarithms, which states [tex]\(\log_b(a^c) = c \log_b(a)\)[/tex]:
[tex]\[ \log_{10}(10^{1000}) = 1000 \cdot \log_{10}(10) \][/tex]
Since [tex]\(\log_{10}(10) = 1\)[/tex]:
[tex]\[ \log_{10}(10^{1000}) = 1000 \][/tex]
3. Calculate the logarithm of the result from step 2:
Now we need to find:
[tex]\[ \log_{10}(1000) \][/tex]
Using the power rule again:
[tex]\[ \log_{10}(1000) = \log_{10}(10^3) \][/tex]
[tex]\[ \log_{10}(10^3) = 3 \cdot \log_{10}(10) \][/tex]
Since [tex]\(\log_{10}(10) = 1\)[/tex]:
[tex]\[ \log_{10}(10^3) = 3 \][/tex]
Therefore, the final result of the expression [tex]\(\log \left(\log \left(10^{1000}\right)\right)\)[/tex] is:
[tex]\[ 3 \][/tex]
So, [tex]\(\boxed{3}\)[/tex].
