Result of log has two component integer part and decimal part, integer part is called characteristic and decimal part is called mantissa.
Suppose log 2 = 0.30103, here 0 is characteristic and 30103 is mantissa.
Now with the help of characteristic we can find the number of digit in a given number.
Like 2 is single digit number i.e (characteristic + 1)
Similarly suppose we want to find number of digit in $2^{10}$ we can calculate it as log $2^{10}$ = 3.0103, Notice here characteristic part is 3, and number of digit will become (characteristic + 1) i.e 4, and $2^{10}$ i.e 1024 is actually 4 digit number.
Similarly we can find number of digit in any cases with the help of characteristic.
Now,number of digit in $2^{56}$ will be log $2^{56}$ = 16.85168, here characteristic is 16, so number of digit will be 17.