  The explicit semiclassical treatment of the logarithmic perturbation theory
for the bound-state problem for the spherical anharmonic oscillator is
developed. Based upon the $\hbar$-expansions and suitable quantization
conditions a new procedure for deriving perturbation expansions is offered.
Avoiding disadvantages of the standard approach, new handy recursion formulae
with the same simple form both for ground and excited states have been
obtained. As an example, the perturbation expansions for the energy eigenvalues
of the quartic anharmonic oscillator are considered.
