Let $\Omega\subset\mathbb C^n$ be a bounded strictly pseudoconvex domain with $C^\infty$ boundary, and let $\rho$ be a smooth defining function with $\rho<0$ on $\Omega$. Near $\partial\Omega$, the diagonal Bergman kernel has an expansion of the form \begin{align*} K_\Omega(z,\bar z)=\frac{\phi(z)}{(-\rho(z))^{n+1}}+\psi(z)\log(-\rho(z)), \end{align*} where $\phi$ and $\psi$ are smooth up to the boundary and $\phi|_{\partial\Omega}$ is nonzero.