# What if we take the logarithm of $X$ ? How does skewness and kurtosis change?

Motivated by this question,

Let $$X$$ be a random variable.

What if we take the logarithm of $$X$$ ? How does skewness and kurtosis change of $$X$$ changes compared to $$\ln{X}$$ ?

• have you tried using the delta method? – Taylor Jul 20 '19 at 4:07
• @Taylor what is delta method ? – onurcanbkts Jul 20 '19 at 7:32
• minutes before the last comment I had just written another answer using it – Taylor Jul 20 '19 at 17:33

Not a complete answer (not yet, anyway) but I thought this should get some response.

The effect of taking logs on skewness and kurtosis depends on the distribution.

Indeed just shifting location will alter the effect of taking logs.

Consider a shifted exponential distribution, for example. With shift parameter 0, the log is left skew, but a small shift will leave the log being right skew, and a larger shift means taking logs doesn't seem to change the shape much: This tells us that the story is not a simple one.

As Taylor points out, Taylor series (/the delta method) can be used to approximate moments of transformed variables, and even their distribution (under some conditions).