I’m trying to calculate the total heat produced by a system over a period of time and I’ve gotten a regression line of y= log x to represent the best produced by the system. To calculate the total heat produced by the system, would it be valid to take the integral of log x? And if this is valid, what would be the units of the area under the curve?
If what you're trying to describe is a thermally isolated* system which is gradually heating up, and if your axes are temperature and time, then the total amount of heat $Q$ produced after a given amount of time is related to the temperature $T$ through the definition of heat capacity:
for a system with heat capacity $C$ and initial temperature $T_0$. Taking the integral will give you an unrelated quantity; the relevant operation is taking the temperature difference between the initial and final times.
This is very similar to how calorimeters measure the amount of energy a reaction produces: a substance with well-known heat capacity absorbs the energy of the reaction, and the temperature difference is measured and plugged into the above formula.
As an aside, I would be careful about drawing any kind of physical meaning from that logarithmic fit. There are a ton of different functions that can be massaged to fit a curve that only has a finite number of data points; without a model to justify the choice of fitting function, there's no reason to believe that this is actually a logarithmic curve rather than a very similar-looking polynomial, rational, or exponential curve. Interpolating between your data points with this fitting function is fine, but without a model, don't trust extrapolation. For example, your logarithmic function supposedly tells you that your system would never stop heating up, no matter how hot it got or how long it was left to run. It also supposedly tells you that at some point in the past, its temperature rapidly declined toward negative infinity, far below absolute zero. In reality, neither of those can be true.
*Without thermal isolation, the rate of heat loss to the environment has to be accounted for; depending on the particular form of this heat loss, the resulting equation could be quite complicated.