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How many digits are in the expansion of 5^30?
How do you do it without a calculator. Show your steps.
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The idea of the solution is to estimate logarithm base 10 of the number N = .
Then use the fact that if log(N) is concluded between two consecutive integer numbers n and (n+1),
n <= log(N) < (n+1), then the integer number N has (n+1) digit.
Without using a calculator, you find from the logarithmic tables
log(5) = 0.69897 (approximately)
(logarithms base 10; for the logarithms table see many Internet sources, for example, this one
http://www.sosmath.com/tables/logtable/logtable.html)
Then = 30*log(5) = 30*0.69897 = 20.969...
It means that the number has 21 digits in base 10 numerical system.
