The most often used additional factor is Pastor-Stambaugh.
The Fama-French model is augmented with a proxy for the Pastor-Stambaugh liquidity factor.
r = RF + βmkt (RM - RF) + βS x SMB + βV x HML + βL x LIQ
You could check the replication issue at Critical Finance
Based on your paper and variables, I assume you ask about the use in econometric models. There are some rules of thumb for taking logs (do not take them for granted). See for example Wooldrigde: Introductory Econometrics P. 46.
When a variable is a positive $ amount, the log is often taken (wages, firm sales, market value...)
Same for variables such as ...
I cite from the fantastic book by Bali, Engle, and Murray (2016): Empirical Asset Pricing: The Cross Section of Stock Returns.
In what follows, they talk about the pricing of size in the stock market (think of market cap = price times shares outstanding). In regressions, you often find researchers using log market cap. Here, the authors explain why:
Generally, we model a lot of quantities in finance as exponentially growing variables like stock price in Black Scholes Model or GDP, because these quantities grow continuously every year. Also, we humans are most comfortable with linear relations, so while studying GDP growth or stock returns, it becomes natural to use logarithms to get linear models.