  This paper is concerned with the random effect of the noise dispersion for
stochastic logarithmic Schr\"odinger equation emerged from the optical fibre
with dispersion management. The well-posedness of the logarithmic Schr\"odinger
equation with white noise dispersion is established via the regularization
energy approximation and a spatial scaling property. For the small noise case,
the effect of the noise dispersion is quantified by the proven large deviation
principle under additional regularity assumptions on the initial datum. As an
application, we show that for the regularized model, the exit from a
neighborhood of the attractor of deterministic equation occurs on a
sufficiently large time scale. Furthermore, the exit time and exit point in the
small noise case, as well as the effect of large noise dispersion, is also
discussed for the stochastic logarithmic Schr\"odinger equation.
