How many digits does the number 2^100 have? Using complicated caculator softwares it is 31 digits but how will you find out with pencil and paper
Hello, sundance240!
How many digits does the number 2^100 have?
Using complicated caculator softwares, it has 31 digits,
but how will you find out with pencil and paper?
We note that: .2^{10} .= .1024 .≈ .10^3
Hence: .2^{100} .≈ .(10^3)^{10} .= .10^{30}
. . which has 31 digits.
In prehistoric times we used to do problems like this by using logarithms to base 10. These were used so often that everybody knew that log(2) = 0.3010... . Therefore log(2^{100}) = 100*log(2) = 30.10..., and since this is between 30 and 31 it follows that 2^{100} is a 31-digit number.