  We use the BLM method to fix the renormalization scale of the QCD coupling in
exclusive hadronic amplitudes such as the pion form factor and the
photon-to-pion transition form factor at large momentum transfer.
Renormalization-scheme-independent commensurate scale relations are established
which connect the hard scattering subprocess amplitudes that control exclusive
processes to other QCD observables such as the heavy quark potential and the
electron-positron annihilation cross section. The commensurate scale relation
connecting the heavy quark potential, as determined from lattice gauge theory,
to the photon-to-pion transition form factor is in excellent agreement with
$\gamma e \to \pi^0 e$ data assuming that the pion distribution amplitude is
close to its asymptotic form $\sqrt{3}f_\pi x(1-x)$. We also reproduce the
scaling and normalization of the $\gamma \gamma \to \pi^+ \pi^-$ data at large
momentum transfer. Because the renormalization scale is small, we argue that
the effective coupling is nearly constant, thus accounting for the nominal
scaling behavior of the data. However, the normalization of the space-like pion
form factor $F_\pi(Q^2)$ obtained from electroproduction experiments is
somewhat higher than that predicted by the corresponding commensurate scale
relation. This discrepancy may be due to systematic errors introduced by the
extrapolation of the $\gamma^* p \to \pi^+ n$ electroproduction data to the
pion pole.
