I will use the same notation I used here: Mathematics behind classification and regression trees
Gini Gain and Information Gain ($IG$) are both impurity based splitting criteria. The only difference is in the impurity function $I$:
- $\textit{Gini}: \mathit{Gini}(E) = 1 - \sum_{j=1}^{c}p_j^2$
- $\textit{Entropy}: H(E) = -\sum_{j=1}^{c}p_j\log p_j$
They actually are particular values of a more general entropy measure (Tsallis' Entropy) parametrized in $\beta$:
$$H_\beta (E) = \frac{1}{\beta-1} \left( 1 - \sum_{j=1}^{c}p_j^\beta \right)
$$
$\textit{Gini}$ is obtained with $\beta = 2$ and $H$ with $\beta \rightarrow 1$.
The log-likelihood, also called $G$-statistic, is a linear transformation of Information Gain:
$$G\text{-statistic} = 2 \cdot |E| \cdot IG$$
Depending on the community (statistics/data mining) people prefer one measure or the the other (Related question here). They might be pretty much equivalent in the decision tree induction process. Log-likelihood might give higher scores to balanced partitions when there are many classes though [Technical Note: Some Properties of Splitting Criteria. Breiman 1996].
Gini Gain can be nicer because it doesn't have logarithms and you can find the closed form for its expected value and variance under random split assumption [Alin Dobra, Johannes Gehrke: Bias Correction in Classification Tree Construction. ICML 2001: 90-97]. It is not as easy for Information Gain (If you are interested, see here).
Best Answer
It depends on what algorithm is being used to build the tree. CART trees are invariant to scale changes so a log transform should not change the resulting tree. However, the values of the split rules will be changed to the log scale.
The reason for this is because the splitting process sorts each feature (numeric) and then checks midpoints between successive observations for impurity improvement based on splits at the interval point. The maximum across observations and features is chosen for that node and the process continues. This means that if you rescale any feature(s) as long as the relative ordering of feature values is maintained-which a log transform will maintain-the tree will be the same but the split values will be log-transformed.