Use a multimeter and have your camera take a picture of the meter periodically. 
Rather than log current, there may be other ways to solve your problem. One might be to measure current with a meter while your project is in various states. Then log the state of the project and infer current from previous measurements. If you are looking to find out energy usage from your current measurements, you could simply time how long your project runs from a known supply of energy (battery, supercap).