The Wikipedia article of the log-normal distribution says
If $X$ is a random variable with a normal distribution, then $Y = \exp(X)$ has a log-normal distribution; likewise, if $Y$ is log-normally distributed, then $X = \log(Y)$ is normally distributed. (This is true regardless of the base of the logarithmic function: if $\log_a(Y)$ is normally distributed, then so is $\log_b(Y)$, for any two positive numbers $a, b ≠ 1$.)
(The general form of this has described the log-normal distribution on Wikipedia since at least 2006.)
I have two questions.
1) is it the case that the logarithm can be taken to any base (i.e., it need not be the natural log), while the exponential must have Euler's number $e$ as the number which is raised to a power? Or, is it the case that I could use other bases to describe the log normally distributed variable, $X = a^{\mu+Z\sigma}$, where $Z$ is a standard normal and $a$ is any positive, real number?
And, conditional on the answer to #1, 2) why the asymmetry?