What is the reason behind taking log transformation of few continuous variables?
This is done when the variables span several orders of magnitude. Income is a typical example: its distribution is "power law", meaning that the vast majority of incomes are small and very few are big. 

This type of "fat tailed" distribution is studied in logarithmic scale because of the mathematical properties of the logarithm:

$$log(x^n)= n log(x)$$

which implies

$$log(10^4) = 4 * log(10)$$

and 

$$log(10^3) = 3 * log(10)$$

which transforms a huge difference $$ 10^4 - 10^3 $$ in a smaller one $$ 4 - 3 $$
Making the values comparable.