  A translation on the d-dimensional torus $\mathbb{T}^d$ has the logarithm
property if the Shrinking Target Property holds for the sequence of balls with
radius $n^{-\frac{1}{d}}$. On $\mathbb{T}^1$ every irrational translations has
this property. In higher dimension, we give criterions based upon an unusual
Diophantine type. In the two dimensional torus, we construct counter examples
to the logarithm with translations vectors of arbitrary small Diophantine type
and example of logarithm property with Liouvillian vector
