  We use the Bernstein method to provide logarithmic gradient estimates and
universal boundedness estimates for semilinear elliptic equations that satisfy
the subcritical index condition. Our estimates are applicable to equations on
Riemannian manifolds with (Bakry-\'{E}mery) Ricci curvature that is bounded
below, and they are original for many concrete equations, even on Euclidean
spaces that have been carefully investigated by numerous experts. Under the
(Bakry-\'{E}mery) Ricci curvature bounded below condition, we establish
relations among basic estimates, such as the universal boundedness estimate,
logarithmic gradient estimate, and Harnack inequality. Under the
(Bakry-\'{E}mery) Ricci curvature non-negative condition and certain mild
assumptions, these estimates are $\kappa$-equivalent for any $\kappa>1$. As an
application, we obtain optimal pointwise estimates and Liouville theorems for
specific equations on Euclidean spaces.
