How many digits does 2 to the 64 have?

Question

The number 2^{64} (the count of values a 64-bit integer can hold) is far too big to write out mentally. But its length is easy. Solve in your head. How many digits does 2^{64} have in base 10? Aim: under 30 seconds.

Accepted Answer

The digit count of a whole number N is \lfloor \log{10} N floor + 1. Use \log{10} 2 \approx 0.301: \log{10} 2^{64} = 64 	imes 0.301 = 19.27. Take the floor and add one: \lfloor 19.27 floor + 1 = 19 + 1 = 20 	ext{ digits.} (The number is about 1.8 	imes 10^{19}, indeed 20 digits long.) Takeaway. To size up any power, multiply the exponent by the base-10 log of the base and read the integer part. With \log{10}2 \approx 0.301 memorised, you can instantly say how long 2^{n} is: 2^{10} has 4 digits, 2^{64} has 20, 2^{100} has 31.

How many digits does 2 to the 64 have?

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