Inaccurate Logarithm in Python

You should read "What Every Computer Scientist Should Know About Floating-Point Arithmetic".

http://docs.sun.com/source/806-3568/ncg_goldberg.html

Always assume that floating point operations will have some error in them and check for equality taking that error into account (either a percentage value like 0.00001% or a fixed value like 0.00000000001). This inaccuracy is a given since not all decimal numbers can be represented in binary with a fixed number of bits precision.

Your particular case is not one of them if Python uses IEEE754 since 31 should be easily representable with even single precision. It's possible however that it loses precision in one of the many steps it takes to calculate log₂2³¹, simply because it doesn't have code to detect special cases like a direct power of two.

This is to be expected with computer arithmetic. It is following particular rules, such as IEEE 754, that probably don't match the math you learned in school.

If this actually matters, use Python's decimal type.

Example:

from decimal import Decimal, Context
ctx = Context(prec=20)
two = Decimal(2)
ctx.divide(ctx.power(two, Decimal(31)).ln(ctx), two.ln(ctx))