  We study a special case of Kashio's $p$-adic $\log\Gamma$-function, that we
call ${\mathrm{Log}}\Gamma_{\mathit{p}}$, which combines these of Morita and
Diamond. It agrees with each of these on large parts of its domain and has the
advantage of being a locally analytic function. We prove a distribution formula
for ${\mathrm{Log}}\Gamma_{\mathit{p}}$ which generalizes and links the known
distribution formulas for Diamond's and Morita's functions.
