Positive real number ($x\in \mathbb{R},x>0$)

A positive real number is a real number $x$ with $x>0$. Zero is excluded; allowing zero gives a nonnegative real number. Both $1/2$ and $\sqrt{2}$ are positive.

The sum, product, and reciprocal of positive real numbers are positive. Consequently the positive real numbers form a group under multiplication, and the natural logarithm converts that multiplication into addition: $\log (xy)=\log x+\log y$. Positivity is also the domain restriction that makes the real logarithm well-defined.