  The trilayer nickelate Nd$_4$Ni$_3$O$_{10-\delta}$ ($\delta \approx$ 0.15)
was investigated by the measurements of x-ray diffraction, electrical
resistivity, magnetic susceptibility, and heat capacity. The crystal structure
data suggest a higher Ni valence in the inner perovskite-like layer. At ambient
pressure the resistivity shows a jump at 162 K, indicating a metal-to-metal
transition (MMT). The MMT is also characterized by a magnetic susceptibility
drop, a sharp specific-heat peak, and an isotropic lattice contraction. Below
$\sim$ 50 K, a resistivity upturn with a log$T$ dependence shows up,
accompanying with a negative thermal expansion. External hydrostatic pressure
suppresses the resistivity jump progressively, coincident with the diminution
of the log$T$ behavior. The low-temperature electronic specific-heat
coefficient is extracted to be $\sim$ 150 mJ K$^{-2}$ mol-fu$^{-1}$, equivalent
to $\sim$ 50 mJ K$^{-2}$ mol-Ni$^{-1}$, indicating an unusual heavy-electron
correlated state. The novel heavy-electron state as well as the logarithmic
temperature dependence of resistivity is explained in terms of the Ni$^{3+}$
centered Kondo effect in the inner layer of the (NdNiO$_3$)$_3$ trilayers.
