Trying to determine if a logarithmic regression analysis is valid. With good R2 values, does it matter that there are ony 7 data points?
In my opinion 7 observations is too small no matter what kind of regression you are running. I would recommend increasing the sample size to get more precise estimates.
It heavily depends on the specification that you are running though, as well as the model. With a logarithmic regression you mean OLS but with the log of the variables I assume. In either case, if you have many regressors and are dealing with a low sample size you'll have problems with low degrees of freedom, translating in a lack of significance. Also if the effect sizes that you are dealing with are small, you'll need more observations.
If you are working with experimental data try to do a power analysis beforehand to get an idea of the sample size needed.
Hope this helps.