How to construct high speed logarithmic amplifiers
Another way to look at it is if you can get the high speed and low sensitivity and IF the device doesn't saturate or slow down if you load it heavily then you need not have to have the log aspect of it.

Look at the Analog devices ADN2880 which is a SiGe based 2.5 GHz Integrated TIA.

And the HMC6590 (which might be too costly for you) also is blazing fast (faster in fact) and can saturate.