Numerous biological systems are known to harbour a form of logarithmic behaviour, from Weber's law to bacterial chemotaxis. Working on a logarithmic scale allows the organism to respond appropriately to large variations in a given input at a modest cost in terms of metabolism. Here we use a genetic algorithm to evolve biochemical networks displaying a direct logarithmic response. Interestingly, a quasi-perfect log-response implemented by the same simple core network evolves in a convergent way across our different replications. The best network is able to fit a logarithm over 4 order of magnitude with an accuracy of the order of 1%. At the heart of this network, we show that a logarithmic approximation may be implemented with one single non-linear interaction, that can be interpreted either as a phosphorylation or as a ligand induced multimerization and provide an analytical explanation of the effect. Biological log-response might thus be easier to implement than usually assumed.