Anti-log or exponential generation is simply a matter of rearranging the logarithmic circuitry. The circuit diagram below shows the circuitry of the log converted to generate an exponential output from a linear input.
The emitter of Q2 in proportion to the input voltage is driven by amplifier A1 in conjunction with transistor Q1. The collector current of Q2 varies exponentially with the emitter-base voltage. [Source: National Semiconductor Application Note]