What is number of digits in the expansion of 125^{100}? (Given \log_{10}2=0\cdot301)

Explanation

Number of digits = \lfloor 100 \log_{10} 125 \rfloor + 1. Note that \log_{10} 125 = \log_{10} (1000/8) = 3 - 3\log_{10} 2 = 3 - 3(0.301) = 2.097. Then 100 \times 2.097 = 209.7. Adding 1 to the integer part yields 210.

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