I'm not sure what kind of Chemistry you're doing at A level, but when I did it there wasn't much application of logic really.
In first year of science, everyone does very similar core modules, which both broadens your horizons in different fields and gets everyone to a base level of understanding.
In biochemistry, you will have to learn basic chemical reactions that are relvant in the human body. In each year, there is at least one module on organic or physical chemistry. However, the rest will be mainly biology based.
I don't expect an A-level student to understand, but the knowledge gained in biochemistry is not simply just "useless information". You are constantly being tested on your ability to rationalize a scenario by applying the information you have learnt, unlike A-level, there are essay and problem questions that require logic in a different sense to pure chemistry, which I think you are more interested in.
For example, in chemistry, you may be given the properties of a mystery compound. And using your knowledge of different reactive groups, you may have to identify, synthesize, or build up a model of this.
In biochemistry, you may be given the properties of a pathogenic molecule. Your goal describe what kind of effects this causes on the body and how it does this in a chemical way, i.e. describe the type of molecules and reactions involved. And possibly how to treat it.
Hence, biochemists help concoct/alter drugs and treatments for doctors to use. Pharmacology applies the knowledge of effect and dose. And Chemistry tries to make new and better compounds by synthesis, using the rules of nature.
I'm currently in my third year now, doing a research project using a machine similar to those used in NMR, but conducts EPR spectroscopy instead that involves a lot of data analysis.
The Biology side generally broader, concepts overlap and are brought together from many different fields.