I think this code is pretty good, I like the docstrings!

Some things strike me as a bit awkward though,

1. Stay DRY

These log functions are really similar, the only thing that differentiates is the operators

You could make the operator an argument of the function, then you'd only have 1 function

You'd then need a lookup table for the operators so you know when to subtract or add.

2. The are better testing modules

Assertions are fine, but are not the best testsuites, there are some good modules for testing in python (I'm a big fan of doctest) but unittest is also pretty good

So I would rewrite the log functions to this:

import operator
import doctest

MATH_OPER = {
                operator.gt: operator.add, 
                operator.ge: operator.add,
                operator.lt: operator.sub,
                operator.le: operator.sub
           }

def log_oper(value, base, op):
    """
    The logarithmic function with an operator
    
    Test all operators
    >>> log_oper(3, 2, operator.gt)
    2
    >>> log_oper(3, 2, operator.ge)
    2
    >>> log_oper(3, 2, operator.lt)
    1
    >>> log_oper(3, 2, operator.le)
    1

    Test assertion Exception
    >>> log_oper(-1, 2, operator.le)
    Traceback (most recent call last):
     ...
    AssertionError: Logarithm is only defined for numbers greater than zero (the power approaches negative infinity as the value approaches zero)
    """
    add_or_sub = MATH_OPER[op]
    assert value > 0, 'Logarithm is only defined for numbers greater than zero (the power approaches negative infinity as the value approaches zero)'
    power, limit = _lut[base][value.bit_length()]
    return add_or_sub(power, op(value, limit))

if __name__ == '__main__':
    max_value = 10**6
    base = 2
    init_upto_bitlen(max_value.bit_length(), base)
    doctest.testmod()

I think this code is pretty good, I like the docstrings!

Some things strike me as a bit awkward though,

1. Stay DRY

These log functions are really similar, the only thing that differentiates is the operators

You could make the operator an argument of the function, then you'd only have 1 function

You'd then need a lookup table for the operators so you know when to subtract or add.

2. The are better testing modules

Assertions are fine, but are not the best testsuites, there are some good modules for testing in python (I'm a big fan of doctest) but unittest is also pretty good

So I would rewrite the log functions to this:

import operator
import doctest

MATH_OPER = {
                operator.gt: operator.add, 
                operator.ge: operator.add,
                operator.lt: operator.sub,
                operator.le: operator.sub
            }

def log_oper(op, value, base):
    """
    The logarithmic function with an operator
    
    Test all operators
    >>> log_oper(operator.gt, 3, 2)
    2
    >>> log_oper(operator.ge, 3, 2)
    2
    >>> log_oper(operator.lt, 3, 2)
    1
    >>> log_oper(operator.le, 3, 2)
    1

    Test assertion Exception
    >>> log_oper(operator.le, -1, 2)
    Traceback (most recent call last):
     ...
    AssertionError: Logarithm is only defined for numbers greater than zero (the power approaches negative infinity as the value approaches zero)
    """
    add_or_sub = MATH_OPER[op]
    assert value > 0, 'Logarithm is only defined for numbers greater than zero (the power approaches negative infinity as the value approaches zero)'
    power, limit = _lut[base][value.bit_length()]
    return add_or_sub(power, op(value, limit))

if __name__ == '__main__':
    max_value = 10**6
    base = 2
    init_upto_bitlen(max_value.bit_length(), base)
    doctest.testmod()