Arbitrariness of Euler's number in exponential of log-normal distribution

Question

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?

Answer 1

The reason that the base doesn't matter is because if $X = \log Y \sim \mathcal N(\mu, \sigma^2)$ then

\begin{align} X_b &= \log_b Y \\ &= \frac{\log Y}{\log b} \\ &= \frac{X}{\log b} \sim \mathcal N\left(\frac{\mu}{\log b}, \left(\frac{\sigma}{\log b}\right)^2\right) \end{align} is normally distributed.

Going the other way, if $Y = e^X \sim \log \mathcal N(\mu, \sigma^2)$, \begin{align} Y_a &= a^X \\ &= e^{(\log a)X} \sim \mathcal \log N\left({\mu}{\log a}, \left({\sigma}{\log a}\right)^2\right) \end{align}

Answer 2

It is just common to use e as the exponent in the negative exponential. Certainly the distribution could be defined using any othe postive number. But the normal distribution is certainly defined in terms of e in the density. The normal density is $\dfrac{1}{\sqrt{2\pi}\sigma}\exp\left[-\frac{(x-\mu)^2}{2\sigma^2}\right]$ where $\mu$ is the mean and $\sigma$ is the standard deviation. The density would have to be written differently if you use a different base. Referring to the lognormal specifically as log base e is also a traditional way to do it though it is not the only way to do it.