  The explicit semiclassical treatment of logarithmic perturbation theory for
the nonrelativistic bound states problem is developed. Based upon
$\hbar$-expansions and suitable quantization conditions a new procedure for
deriving perturbation expansions for the one-dimensional anharmonic oscillator
is offered. Avoiding disadvantages of the standard approach, new handy
recursion formulae with the same simple form both for ground and exited states
have been obtained. As an example, the perturbation expansions for the energy
eigenvalues of the harmonic oscillator perturbed by $\lambda x^{6}$ are
considered.
