x = 1

{x element R : x>0} (all positive real numbers)

R (all real numbers)

bijective from its domain to R

d/dx(log(x)) = 1/x

integral log(x) dx = x (log(x) - 1) + constant (assuming a complex-valued logarithm)

log(x) = log(e, x)

log(x) = log(a) log(a, x)

log(x) = -Li_1(1 - x)

log(x) = - sum_(k=1)^∞ ((-1)^k (-1 + x)^k)/k for abs(-1 + x)<1

log(x) = log(-1 + x) - sum_(k=1)^∞ ((-1)^k (-1 + x)^(-k))/k for abs(-1 + x)>1

log(x) = 2 i π floor(arg(x - ξ)/(2 π)) + log(ξ) - sum_(k=1)^∞ ((-1)^k (x - ξ)^k ξ^(-k))/k for ξ<0

log(x) = integral_1^x 1/t dt

log(x) = -i/(2 π) integral_(-i ∞ + γ)^(i ∞ + γ) ((-1 + x)^(-s) Γ(-s)^2 Γ(1 + s))/Γ(1 - s) ds for (-1<γ<0 and abs(arg(-1 + x))<π)

integral_0^1 log(x) dx = -1