Extragalactic studies have demonstrated there is a moderately tight ($approx$0.3 dex) relationship between galaxy stellar mass ($M_{star}$) and star formation rate (SFR) that holds for star-forming galaxies at $M_{star} sim 3 times 10^8$-10$^{11}~M_{odot}$, i.e., the star formation main sequence. However, it has yet to be determined whether such a relationship extends to even lower mass galaxies, particularly at intermediate or higher redshifts. We present new results using observations for 714 narrowband H$alpha$-selected galaxies with stellar masses between $10^6$ and $10^{10}~M_{odot}$ (average of $10^{8.2}~M_{odot}$) at $z approx$ 0.07-0.5. These galaxies have sensitive UV to near-infrared photometric measurements and optical spectroscopy. The latter allows us to correct our H$alpha$ SFRs for dust attenuation using Balmer decrements. Our study reveals: (1) for low-SFR galaxies, our H$alpha$ SFRs systematically underpredict compared to FUV measurements, consistent with other studies; (2) at a given stellar mass ($approx $10$ ^{8}~M_{odot}$), log(specific SFR) evolves as $ A log(1+z) $ with $ A = 5.26 pm 0.75 $, and on average, specific SFR increases with decreasing stellar mass; (3) the SFR-$M_{star}$ relation holds for galaxies down to $sim$10$^6~M_{odot}$ ($sim$1.5 dex below previous studies), and over lookback times of up to 5 Gyr, follows a redshift-dependent relation of $log{({rm SFR})} propto alpha log(M_{star}/M_{odot}) + beta z$ with $alpha = 0.60 pm 0.01$ and $beta = 1.86 pm 0.07$; and (4) the observed dispersion in the SFR-$M_{star}$ relation at low stellar masses is $approx$0.3 dex. Accounting for survey selection effects using simulated galaxies, we estimate the true dispersion is $approx$0.5 dex.
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