I’m working on an ebook edition of a novel by Sinclair Lewis, Arrowsmith, published in 1925, based on scans available at the Internet Archive. At one point, a fictional paper is quoted, with an equation:
In a preliminary publication, I have reported a marked qualitative destructive effect of the radiations from radium emanations on Bacteriophage-anti-Shiga. In the present paper it is shown that X-rays, gamma rays, and beta rays produce identical inactivating effects on this bacteriophage. Furthermore, a quantitative relation is demonstrated to exist between this inactivation and the radiations that produce it. The results obtained from this quantitative study permit the statement that the percentage of inactivation, as measured by determining the units of bacteriophage remaining after irradiation by gamma and beta rays of a suspension of fixed virulence, is a function of the two variables, millicuries and hours. The following equation accounts quantitatively for the experimental results obtained:
\begin{equation}
K = \frac{λ \log e \frac{u_0}{u}}{E_0 (ε-λt_1)}
\end{equation}
The ebook needs to represent this equation in MathML, which means I need to know which adjacent symbols are being multiplied and which are function applications. But the logarithm in the numerator seems ambiguous. It could be:
\begin{equation}
\log(e) \times \frac{u_0}{u}
\end{equation}
or:
\begin{equation}
\log\left(e \times \frac{u_0}{u}\right)
\end{equation}
or:
\begin{equation}
\log_e\left(\frac{u_0}{u}\right)
\end{equation}
Given the context, that this equation describes the fictional decline of a population based on two variables, which, if any, of these three possibilities is more likely than the others?
I’ve checked several modern ebook editions, but besides having execrable mathematics formatting, they retain the same ambiguity.
Best Answer
Note that in the original typeset equation you have linked to, there is a barely visible underline _ beneath the $e$ which is almost obscured by the dividing line. This obviously isn’t meant to be there. Clearly the original typesetter had no problems with setting subscripts as these are present elsewhere in the formula. Therefore I guess that what happened is that whoever was proofreading the text may have indicated that the $e$ was meant to be a subscript, but they used the wrong proof-reading symbol, and so the typesetter merely incorporated the underline into the formula thinking it was meant to be there. And no one checked it before printing.
Without knowing the actual meaning of the various letters in the formula, the following is therefore also guesswork:
The first option you suggest is clearly wrong since log e is 1 and no self-respecting scientist would leave their formula in this unsimplified form.
The second option is highly unlikely since the numerator would then be $$\lambda\left(1+\log\left(\frac{u}{u_0}\right)\right)$$ and it is difficult to see how the extra $1$ would have arisen in the formula.
The third option is highly likely to be correct. It was, and still is, conventional to use just the word $\log$ to denote natural log without risk of ambiguity, but also conventional to write $\log_e$ to avoid any possibility of confusion with $\log_{10}$.
If you think it’s necessary to change it from the ambiguous original then I would suggest option 3.
I hope this helps.