section_path: Complex analysis/Complex logarithm
section_name: Complex logarithm
paragraph_text: Complex logarithm
  
  For any positive real number t, there is a unique real number x such that exp (x) = t {\displaystyle \exp(x)=t}. This leads to the definition of the natural logarithm as the inverse ln: R + → R; x ↦ ln x {\displaystyle \ln \colon \mathbb {R} ^{+}\to \mathbb {R};x\mapsto \ln x} of the exponential function. The situation is different for complex numbers, since
  
  by the functional equation and Euler's identity. For example, e = e = −1, so both iπ and 3 iπ are possible values for the complex logarithm of −1.
  
  In general, given any non-zero complex number w, any number z solving the equation
  
  is called a complex logarithm of w, denoted log w {\displaystyle \log w}. It can be shown that these numbers satisfy z = log w = ln | w | + i arg w, {\displaystyle z=\log w=\ln |w|+i\arg w,} where arg is the argument defined above, and ln the (real) natural logarithm. As arg is a multivalued function, unique only up to a multiple of 2 π, log is also multivalued. The principal value of log is often taken by restricting the imaginary part to the interval (− π, π ]. This leads to the complex logarithm being a bijective function taking values in the strip R + + i (− π, π ] {\displaystyle \mathbb {R} ^{+}+\;i\,\left(-\pi,\pi \right]} (that is denoted S 0 {\displaystyle S_{0}} in the above illustration) ln: C × → R + + i (− π, π ]. {\displaystyle \ln \colon \;\mathbb {C} ^{\times }\;\to \;\;\;\mathbb {R} ^{+}+\;i\,\left(-\pi,\pi \right].}
  
  If z ∈ C ∖ (− R ≥ 0) {\displaystyle z\in \mathbb {C} \setminus \left(-\mathbb {R} _{\geq 0}\right)} is not a non-positive real number (a positive or a non-real number), the resulting principal value of the complex logarithm is obtained with − π < φ < π. It is an analytic function outside the negative real numbers, but it cannot be prolongated to a function that is continuous at any negative real number z ∈ − R + {\displaystyle z\in -\mathbb {R} ^{+}}, where the principal value is ln z = ln(− z) + iπ.
  
  Complex exponentiation z is defined as z ω = exp (ω ln z), {\displaystyle z^{\omega }=\exp(\omega \ln z),} and is multi-valued, except when ω is an integer. For ω = 1 / n, for some natural number n, this recovers the non-uniqueness of n th roots mentioned above. If z > 0 is real (and ω an arbitrary complex number), one has a preferred choice of ln x {\displaystyle \ln x}, the real logarithm, which can be used to define a preferred exponential function.
  
  Complex numbers, unlike real numbers, do not in general satisfy the unmodified power and logarithm identities, particularly when naïvely treated as single-valued functions; see failure of power and logarithm identities. For example, they do not satisfy a b c = (a b) c. {\displaystyle a^{bc}=\left(a^{b}\right)^{c}.} Both sides of the equation are multivalued by the definition of complex exponentiation given here, and the values on the left are a subset of those on the right.