In the logarithmic representation, multiplication becomes addition. I don't think that just by doing everything in logarithms you really increase the overall complexity, rather you transfer it from multiplication onto addition, so I don't think adding logarithms should be more expensive than integer multiplication..
Should it then matter for neural networks, which IIRC require similar amount of additions and multiplications?
Even floating-point is actually trading off the simplicity of addition in order to make multiplication easier, because they are quasi-logarithmic representation.
However, I wonder if there is a numeric representation where the circuits for addition and multiplication are of similar complexity. Something like half-logarithm, which if applied twice, you get the logarithm.