Limiting Output Voltage of the Opamp-to avoid saturation
First, you could try replacing the diodes with square-law device tied BJT's.  As it mentions in the image, the stability improves.



Instead of reinventing the wheel, it may be useful to check out TI's guide to logarithmic amplifiers.  In particular, figure 1 with I_in as the input current. 



To quote the pdf a bit more ... "This confirms that the circuit output voltage is logarithmically related to the circuit's input."