A simple upper bound:

>! 7^2 = 49 is just under 50, so 7^100 = (7^2)^50 is somewhat less than 50^50 = (100/2)^50.

>! 2^10 = 1024 is just over 1000 = 10^3, so 2^50 is somewhat over 10^15.

>! So 7^50 is less than (100^50) / (10^15) = ((10^2)^50) / (10^15) = (10^100) / (10^15) = 10^85 (1E85).

And a semi-simple lower bound:

>! 7^6 = 117649 is somewhat more than 10^5, so 7^96 = (7^6)^16 is somewhat more than (10^5)^16 = 10^80.

>! 7^100 = (7^96) * (7^4) is somewhat more than 10^80 * 2401 = 2.401E83.

Or, borrowing an answer from the previous problem: If you have

>! a log table giving log10(7) ~= 0.845

then you can determine that

>! log10(7^100) = 100 * log10(7) ~= 84.5

which produces an estimate of

>! sqrt(10) * 10^84 ~= 3.162E84