Sometimes data is somewhat scaled by a monotonic transformation such as a logarithm or square root for positive real values. It keeps the order of the values intact and hence results obtained with this transformed variable interpretable.

The desired effects of this transformation could be a scaling of the regression coefficients or improving their quality amongst others.

In the provided paper, my guess is that a rescaling of the book-to-market ratio is intended.

Sometimes data is somewhat scaled by a monotonic transformation such as a logarithm or square root for positive real values. It keeps the order of the values intact and hence results obtained with this transformed variable interpretable.

The desired effects of this transformation could be a scaling of the regression coefficients or improving their quality amongst others.

In the provided paper, my guess is that a rescaling of the book-to-market ratio is intended.

Another reason is that the variable may be highly skewed to the right, which may produce problems in a regression analys. By taking the log, the variables tend to behave more in line with the normatlity assumption. It also is a good way of dealing with outliers.

but why we take log of ratio? because ratio is already in normal form..

By taking log; the distribution is more likely to behave like Normal distribution and hence provide better regression analysis. Book-to-market ratio is always positive.

But, log of the ratio could take values that are positive as well as negative.

Right Krishna... yeah before taking log ratios were positive but after taking log they become negative......

Statistical aspects are discussed under this link:

http://mathbabe.org/2011/08/30/why-log-returns/

To reduce the sharpness of the data one can use smoothing only if you are not interested in the standard deviation of the variable.

1) Returns are often assumed to be distributed normally. It's a matter of tolerance whether that is acceptable in a given case. In case of stock returns the bell curve is quite an idealization due to skewness, fat tails and "leptocurtic" peaks. However, when one wants to use the Black-Scholes formula there is no way around that assumption.

2) Since returns are defined as  r = log nat(St / St-1) the underlying S must be assumed to be lognormally distributed.

By taking log; the distribution is more likely to behave like Normal distribution and hence provide better regression analysis. Book-to-market ratio is always positive.

The many comments that book-to-market (/market-to-book) ratios are always positive are not correct. While the stock price can't be negative the bookvalue very well can be when liabilities (& intangible assets) outweigh total assets. A firm that has a sustained string of negative earnings reports may eventually end up with negative book value.

See also:

Ching-Lih Jan and Jane A. Ou (2012) Negative-Book-Value Firms and Their Valuation. Accounting Horizons: March 2012, Vol. 26, No. 1, pp. 91-110.

https://doi.org/10.2308/acch-50094

https://economics.stackexchange.com/questions/17718/log-of-a-negative-market-book-ratio-in-a-regression-model