$log_a$(xy) = $log_a$(x) + $log_a$(y), log_a$$\frac{x}{y} = $log_a$(x) - $log_a$(y), $log_a$($x^n$) = n*$log_a$(x). These hold only when all arguments are positive. Common error: writing log($x^2$) = 2log(x) without noting that log($x^2$) is defined for all x != 0, but 2log(x) requires x > 0. In JEE, this distinction matters when the original expression has $x^2$ inside the log.