A real number x has a log-Exponential distribution if y = exp(x) has Exponential distribution.
The log-Exponential distribution is defined over real numbers. Saying that x is log-Exponential is equivalent to saying that y = exp(x) is Exponential. The log-Exponential distribution therefore expresses lack of information about the order of magnitude of a scale parameter: if x has a log-Exponential distribution, then it has equal chance to be contained by any of the intervals of the form (10^k, 10^(k+1)) within the allowed range.
The density is p(x) = ???, which can be seen by defining x = ln(y) where y has Exponential distribution and apply the change-of-variable formula.
# a log-Exponential prior over the rate of change of a Brownian trait (or a Brownian relaxed clock)
trueTree = readTrees("data/primates.tree")
log_sigma ~ dnLogExponential(lambda=1)
sigma := exp(log_sigma)
X ~ dnBrownian(trueTree,sigma)