There are generally speaking and as far as I can tell 3 options that would work - but only the actual log amp will result in a continuous, analog signal:
1: A log amp. Instead of making one from scratch, I'd opt for one of the integrated types available that use a minimum of external components and offer inherently better temperature compensation than whatever you can cook up at home. Several manufacturers still have a product offering to suit, sometimes even quite recently introduced types.
2: The crude digital approach would be to take an A/D converter of sufficiently high resolution to get the range you need. In practice this means 16 bit+ with overall excellent linearity, low noise etc.
3: A programmable gain linear amplifier will also work, but you need to have some kind of 'auto ranging' logic that determines which gain to set. Then feed the output into an ADC and do the math.
Evidently, 2 & 3 are only useful in systems where you need to have a digital signal. Although arguably, you could do the log & linearization part in the digital domain and then output the linear signal through a DAC (or PWM through an RC filter) to whatever circuitry that wants an analog input.
I'm not aware of any true analog alternative to #1. Log amp IC's are relatively expensive; a quick search suggests that something like
TL441 is a reasonable compromise between cost, performance and availability.
Anyway, back to the main question - what kind of output do you need over what kind of range?