  A simple uniform approximation of the logarithmic derivative of the ground
state eigenfunction for both the quantum-mechanical anharmonic oscillator and
the double-well potential given by $V= m^2 x^2+g x^4$ at arbitrary $g \geq 0$
for $m^2>0$ and $m^2<0$, respectively, is presented. It is shown that if this
approximation is taken as unperturbed problem it leads to an extremely fast
convergent perturbation theory.
