Sending a real number over a noisy channel with limited energy budget
The terms of the kind $N(0, 1)*\alpha/E$ are simply representing noise to the system and the way one defines the multiplicative factor $\alpha/E$ simply changes the variance of the gaussian $N(0,1)$ without impacting the mean. 

Note: the mean of $R(E)$ is $X$

So the distribution of $R(E) = X + N(0,1)\alpha/E$ is $N(X,\frac{\alpha^2}{E^2})$ and for the received $R(E) = X + N(0,1)\alpha ln(E)$ is $N(X,\alpha^2ln(E)^2)$.

In the first case as the energy of the signal is increasing the effect of noise is decreasing (it's variance). So it makes more sense compared to second one where the variance on noise is increasing with increasing signal energy. However it does not reflect the true SNR

Edit: The correct way with the update in your question is $\sqrt(E)$ term, this is because this truly represnts the actual SNR, and this is what should be used to decide the minimum error detection rule