The value of log⁡ab−log⁡|b|=

log ab is defined if ab > 0 or a and b have the same sign. Case (i): a, b > 0 ⇒ log⁡ab−log⁡|b|=log⁡a+log⁡b−log⁡b=log⁡a----iCase (i): a, b < 0⇒log⁡ab−log⁡|b|=log⁡(−a)+log⁡(−b)−log⁡(−b)=log⁡(−a)---iiFrom Eqs. (i) and (ii), we have log⁡ab−log⁡|b|=log⁡|a|