  The efficient and simple B-splines variational method of calculating the
hydrogen atom Bethe logarithms in the acceleration gauge [Y-B Tang et.al.,
Phys. Rev. A $\mathbf{87}$, 022510 (2013)] is successfully applied to other
gauges. The ground state Bethe logarithm of H with fourteen accurate figures is
obtained in the velocity gauge, and in the length gauge the ground state value
has eleven accurate figures. Present velocity- and length-gauge results for the
$ns$, $np$, $nd$, and $nf$ states up to $n = 200$ of H are at the 10$^{-10}$
level of accuracy. This represents the successful variational attempt to
calculate Bethe logarithms of the hydrogen atom in the velocity and length
gauges. Furthermore, the B-splines variational method is successfully extended
to the helium atom by the aid of configuration interaction. Bethe logarithms
for $n\,^{1,3}S\,\,(n=1-8)$ states are calculated in three different gauges.
The acceleration-gauge value of 4.364 036 7(2) a.u. and the
velocity-acceleration gauge value of 4.364 036 4(2) a.u. for the $2\,^3S$ state
are achieved, which are both at the 10$^{-7}$ level of accuracy. For other
triplet states, present results in three different gauges all have five to
seven accurate figures. While for the singlet states, the best convergent
values are obtained in the pa-gauge, of which the numerical precision is also
at the $10^{-5}$ to $10^{-7}$ level of accuracy.
