# Amount of numbers in an exponent

• Nov 10th 2011, 02:04 AM
ienavolante
Amount of numbers in an exponent
sorry for my english and sorry if it's not the right forum(Nod)

i'm asked to say the amount of numbers of 2^56, and I don't know how!
thank u!!
• Nov 10th 2011, 06:54 AM
Deveno
Re: Amount of numbers in an exponent
do you mean the number of digits?

if so we want to find an integer x so that 10^(x) < 2^56 < 10^(x+1).

taking common logarithms: x < log(2^56) < x+1

x < 56(log(2)) < x+1

log(2) is approx. 0.301, so 56(0.301) = 16.856....therefore 2^56 has 17 digits.

(10^x has x+1 digits, a 1 and x 0's after)
• Nov 10th 2011, 07:19 AM
ienavolante
Re: Amount of numbers in an exponent
thank you, yes I meant it!

but i still don't get why 2^56 has to be included between 10^x and 10^(x+1)
• Nov 10th 2011, 09:20 AM
Deveno
Re: Amount of numbers in an exponent
10^x is the smallest number with x+1 digits, and 10^(x+1) is the smallest number with x+2 digits.

for example: 10 is the smallest number with 2 digits (9 has just one digit), and 100 is the smallest number with 3 digits (99 has only 2).

so if a number has 2 digits, it lies between 10 and 100.
• Nov 11th 2011, 02:06 AM
ienavolante
Re: Amount of numbers in an exponent
thank you for the clear explanation!!