They are not cheap to compute, I believe the best way to approximate them is using expensive polynomial approximation which consumes a few cycles and operations.

All they do is map the numbers into a different range. A logarithmic value can represent more numbers near 0, but in order to add two logarithms, they need to be converted back to normal form, where the small numbers are rounded down to zero again.

There's no way around that, as adding a very small number to a very large number will always require many bits of precision to do accurately, regardless what transforms you use on the numbers.