Logarithmic amplifier with high slew rate -- mutually exclusive?
Progressive-limiting Log amps might satisfy. Log conformance for many of these is +/- 0.5dB and accommodate a 60dB dynamic range.

Since you specify an input voltage source, most accept input voltage, and yield an output that is also voltage, in the ballpark of 20 mV/dB.
A big problem is output bandwidth. The fastest I see (AD8310) has rise time of 15 ns, delayed by 6 ns. However, bandwidth is not dependent on amplitude.

I take it that the occasional input pulse response above a somewhat noisy DC background is of most interest. A low-frequency offset compensation loop works in your favor here. To achieve a good output noise floor and good dynamic range, you might want to consider a high-pass filter at the input.

These log-converters respond equally to negative-going pulses as well as positive-going pulses. This non-linear response doesn't work in your favor - should your unipolar pulse overshoot, the negative-going overshoot is converted to a positive-going tail.