Algebraic properties of logarithms

ln(xy) = ln x + ln y
ln(x^{y}) = y ln x
ln(1/x) = − ln x

ln x is defined only for x > 0. ln 1 = 0 ; ln x < 0 if and only if 0 < x < 1.

ln e = 1 ; log_{a}(a) = 1

Except as noted in the last line, these laws also hold for log_{a}.