Exponent of Uniform Distribution

Question

Let $u \in U[a, b]$ - the uniform distribution. Is there a name for the distribution of random variable $c^u$ for both, discrete and continuous cases?

Answer 1

If $u$ is uniformly distributed and $c, a, b>0$ are constants, then $c^u$ has a log-uniform or reciprocal distribution.

Note that the parameters $a, b$ (as given on the Wikipedia page) are for the special case that $c=e$ and they are different from the $a$ and $b$ that you use, but any $c>0$ gives rise to a log-uniform distribution.