Convert ln(mm^2) to linear cm^2

Question

I'm working on a meta-analysis of data extracted from scientific literature. One paper reported their data as $ln(mm^2)$ (see panel F in the figure below from https://doi.org/10.1007/s11258-009-9695-z). I need to convert this to raw $cm^2$ as they were originally measured.

How should I do this conversion?

Answer 1

It does not make sense to take $\ln$ of a dimension; i.e. the argument of $\ln$ should be dimensionless. That said, if we are considering the values $y = \ln \left( \frac{x}{1 \text{ mm}^2} \right)$, then note that $x = (1 \text{ mm}^2) e^y$. And since $1 \text{ mm}^2 = 10^{-2} \text{ cm}^2$, then $x = (10^{-2} \text{ cm}^2) e^y$.

Answer 2

Let $A_m$ be the area in $mm^2$ and $A_c$ be the area in $cm^2$. We have $A_c = A_m / 100$.

We want a formula for $A_c$ given $x = ln(A_m)$. We have $e^x= A_m$, so $A_c = A_m / 100 = \frac{e^x}{100}$.