The obvious question is what's a 'Shooting Log'? Something you lean on as a rest when taking the shot?
Seriously though, have they made this request to you in writing and given their justification for it? It would then be a relatively simple matter to challenge it if it is a 'blanket policy'. What force and how long have you had the FAC, any other restrictions on it? Maybe someone from the same force area can say if this is something that is being applied across the board or just the idea of an over-zealous FEO and applied to yourself.
The HO Guidance says regarding Territorial Conditions at Ch 10. 35 (a):
When a chief
officer is satisfied that a certificate holder has
gained sufficient experience with a particular
calibre or class of firearm the less restrictive
condition may be considered appropriate.
A shooting log hardly makes any sense to provide evidence of experience though, unless it is varified and witnessed by someone they nominate, due to absolutely anything being able to be recorded in it - 'shot 20 deer today' etc. etc.