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The logit link gives ${\mu }_i=e^{\beta x_i}/(1+e^{\beta x_i})$. Independence in the Bernoulli logistic-regression model therefore gives the log-likelihoodDifferentiate using $d{\mu }_i/d\beta =x_i{\mu }_i(1-{\mu }_i)$:The maximum-likelihood estimator, if finite, solves the single score equation ${\sum }_i{x}_i(y_i-{\hat{\mu }}_i)=0$. If at least one $x_i$ is nonzero, the log likelihood is strictly concave at finite $\beta$, so any root is the unique maximizer.

Use Newton method, equivalently one-parameter iteratively reweighted fitting:A damped step or a bracketed score solver is useful if a full step overshoots. The finite maximum-likelihood estimate in a one-parameter logistic model requires the score limit at minus infinity to be positive and its limit at plus infinity to be negative. Perfect sign separation of the observations can instead send the estimate to an infinite coefficient; if every $x_i=0$, the coefficient is unidentifiable. These possibilities should be checked before claiming convergence to a finite estimate.