How do I measure ratio of EEG frequency band for a few electrodes in decibels? I wanted to calculate the ratio of Theta/Beta ($\Theta$/$\beta$) in decibels for a couple of electrodes in AF3 AF4 and Cz

I was able to get these in microvolts, but would like to convert them to decibels but am unsure what is the appropriate way to do so.

For instance, should I convert it like:

$$ratio_1 = 10.log\left( \frac{\Theta_{AF3} + \Theta_{AF4} + \Theta_{Cz}}{\beta_{AF3} + \beta_{AF4} + \beta_{Cz}}\right)$$

or

$$ratio_2 = 10.log\left(\frac{\Theta_{AF3}}{\beta_{AF3}}\right) + 10.log\left(\frac{\Theta_{AF4}}{\beta_{AF4}}\right) + 10.log\left(\frac{\Theta_{Cz}}{\beta_{Cz}}\right)$$
Decidels are ratios in log scale. Usually the ratio of powers (in the physical sense). Powers sum up, but not dB. For example, a sound source at 70dB SPL means its power is approximately 3200 times higher than the reference sound source of dB SPL. If you have 2 sound sources both at 70dB, their power add up but their total sound pressure is going to be only 76dB SPL.

The appropriate way to answer your question depends on the assumptions you make. Your first equation compares the overall power within the Theta frequency band to the overall power within the Beta band, across recording locations. Your second equation compares power ratios at each location independently (although I would take the mean, not add them). Whether the first or the second equation is more appropriate is a scientific question.

Also note that dBs are usually defined in log10 scale ($\frac{ln(x)}{ln(10)}$), but it doesn't really matter.