How many digits does 2 to the 1000 have?

Question

How many digits does 2 to the power 1000 have when written out in decimal? You may use the fact that the logarithm of 2 to base 10 is about 0.30103. Then estimate the first digit, and explain why the rough rule "2 to the 10 is about 1,000" gives a slightly wrong digit count here.

Accepted Answer

302 digits, and it begins with 1. The count. The base-10 logarithm of 2 to the 1000 is 1000 × log 2 = 1000 × 0.30103 = 301.03. So 2 to the 1000 is 10 to the 301.03, which lies between 10 to the 301 and 10 to the 302. A number in that range has 302 digits (10 to the 301 itself is a 1 followed by 301 zeros, 302 digits). The first digit. 10 to the 301.03 = 10 to the 301 × 10 to the 0.03, and 10 to the 0.03 is about 1.07. So 2 to the 1000 is about 1.07 × 10 to the 301: it starts with the digits 107. (The exact value begins 10715086...) Without the logarithm table. The rough rule "2 to the 10 is a…

How many digits does 2 to the 1000 have?

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