For log converters you should not use diodes, because their parasitic currents mask the current component that has an exponential dependence from the voltage.

For log converters, bipolar transistors are used, because their collector current depends only on the ideal diode current of the base-emitter diode, not also on its parasitic currents, so the base-emitter voltage has a logarithmic dependence on the collector current, for a relatively wide range of currents.