This is something that you cannot do; there is no damaging option to Control Flames, nor does one increase the damage or keep a fire from being put out.
Beware weapon breakage if you go this route; burning logs are not known for their stability.
Also beware the fire damage you will be taking for this.
So. Um. Why are you not taking damage from being set on fire, again? Nothing in the spell prevents or reduces fire damage.
How are you reducing the rate of burning? Not even reduction in light emitted reduces burn rate.