Accurately determining the time constant of a circuit enables IoT nodes to easily read out resistive or capacitive sensors. However, power and cost constraints lead to hardware limitations that complicate such measurements, including ADC noise, sampling clock jitter, poor voltage control over temperature and process, and a low power microprocessor without a fast multiplier or floating point support. This work discusses estimating the time constant of a decaying exponential's ADC samples using a simple weighted linear regression and describes the on-chip implementation of the regression on a low-cost, low-power microprocessor. Experimental results with an imperfect ADC show that time constants over more than two orders of magnitude can be accurately estimated within 5% of the nominal value with a mean standard error of about 1% of the nominal value.