What is the distribution of the pKa from the Henderson-Hasselbalch equation given log-normal concentrations? Background
The Henderson-Hasselbalch equation is given by
$$\text{pH} = \text{pK}_{\text{a}} + \log_{10} \left( \frac{[\text{Base}]}{[\text{Acid}]} \right)$$
where $\text{pH} = -\log [\text{H}^+]$. Solving for the $\text{pK}_{\text{a}}$ we obtain the equality
$$\text{pK}_{\text{a}} = \text{pH} - \log_{10} \left( \frac{[\text{Base}]}{[\text{Acid}]} \right).$$
Just to make our notation simpler, let us substitute:

*

*$W :=\text{pK}_{\text{a}}$

*$X := [\text{H}^+]$

*$Y := [\text{Acid}]$

*$Z := [\text{Base}]$
giving us the expression
$$W = - \log_{10}X  - \log_{10} \left( \frac{Z}{Y} \right)$$
which is equivalent to
$$W = \log_{10} \left( \frac{Y}{XZ} \right).$$
Those familiar with using the equation may raise an eyebrow at treating $\text{pK}_{\text{a}}$ as a random variable since it is often taken to be a constant. Here I am supposing that variability enters into our calculation of the $\text{pK}_{\text{a}}$ through the variables $X,Y,Z$ that are themselves not known with perfect precision.
Assume that $X,Y,Z$ have a joint log-normal distribution. Statistical independence cannot be assumed.
Question
What is the distribution of $W$?

$W$ is Normal:
$$W=-\log_{10} X -\log_{10} Z+\log_{10} Y$$
and by assumption the three quantities on the right-hand side are multivariate Normal, so their sum is Normal.
