Do all order of moments exist for the log-exponential family (both multivariate and single-variate)? I already know they exists for log-normal distribution: single variate $EX^n=e^{n\mu+\frac{n^2\sigma^2}{2}}$ and multivariate $E(X\_1^{k\_1}\cdots X\_n^{k\_n})=e^{k'\mu+\frac{1}{2}k'\Sigma k}$. 

If not, how about the exponential family without log?

How about log-normal-mixture or or log-RBF-kernel-density or RBF-kernel density estimate without log?