How many digits does the number 125^100 have? Im breaking my head here :(
100*log10(125) ~ 210?
(We can observe that 125 = 1000/8, so 100log10(125) = 300 - 300log10(2) = 300 - 30log(1024). From here we merely need the bound 90 = 30log(1000) < 30log(1024) < 91, or equivalently 1 < 1.024^30 < 10. There are probably better approaches but I'm partial to 1.024^30 = (1.024^15)^2 < (1.024^16)^2 < 3^2 < 10, since 1.024^16 is easy to upper bound by repeated squaring and rounding up: 1.024^16 < 1.03^16 < 1.1^8 < 1.3^4 < 1.7^2 < 2.9.)