Find the leftmost (most significant digits) of a large exponent calculation, say 99^(99) I want to find the initial 10 digits of an exponent calculation whose result is a very large number - Say, 99^(99) = 3.697296 xx 10^(197)

bucstar11n0h

bucstar11n0h

Answered question

2022-11-19

Find the leftmost (most significant digits) of a large exponent calculation, say 99 99
I want to find the initial 10 digits of an exponent calculation whose result is a very large number - Say, 99 99 = 3.697296 × 10 197
I only need to know the digits 3697296
Is there any way of finding this without doing the full calculation?

Answer & Explanation

kavdawg8w8

kavdawg8w8

Beginner2022-11-20Added 20 answers

Use common logarithms and a decent calculator. Using a fairly mediocre one on your example, I get 99 log 10 99 197.5678842652; subtracting 197 and raising 10 to the resulting power, I get 3.697296376497, so the number must be about 3.697296376497 × 10 197
Added: More generally, for a b calculate b log 10 a, subtract the integer part, and raise 10 to the resulting power. If n = b log 10 a , the integer part of b log 10 a, your number is
10 b log 10 a n × 10 n ,
and you’ll be able to read of the most significant digits from the lefthand end of 10 b log 10 a n .

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