Logarithms and Complex Powers

The complex logarithm of z ≠ 0 is ln z = ln|z| + i(Arg z + 2kπ), so it has infinitely many values that differ by multiples of 2πi, and complex powers are defined by z^w = e^{w ln z}. In brief: ln(−1) = iπ + 2kπi, ln i = iπ/2 + 2kπi and iⁱ = e^{i · iπ/2} = e^−π/2 ≈ 0.2079. The principal value Ln z uses Arg z in (−π, π].

The real logarithm is defined only for positive numbers. In the complex domain, the logarithm exists for every nonzero number, but it is multi-valued, because e^z is periodic with the imaginary period 2πi. This leads to surprising results, such as the fact that iⁱ is a real number. This guide defines the complex logarithm and the principal value, explains the rules and their limits, shows how to compute complex powers, covers logarithms of negative numbers and gives a table of values.

How is the complex logarithm defined?

A logarithm of z is any number w with e^w = z. Write z = re^iθ and w = u + iv: e^ue^iv = re^iθ, so e^u = r and v = θ + 2kπ. Therefore

ln z = ln|z| + i(Arg z + 2kπ), k an integer.

The principal value, Ln z, takes k = 0: Ln z = ln|z| + i Arg z, with the principal argument −π < Arg z ≤ π. The principal logarithm is discontinuous on the negative real axis, where the argument jumps from π to −π. The real logarithm is the special case for positive z, where Arg z = 0.

This extends the real logarithm to negative numbers: Ln(−5) = ln 5 + iπ ≈ 1.6094 + 3.1416i. The statement that “the logarithm of a negative number does not exist” is true only for the real logarithm.

Which rules hold for complex logarithms?

• e^{ln z} = z for every value of ln z.
• ln(zw) = ln z + ln w holds as an equality of sets of values (all values on both sides), but for principal values only up to a multiple of 2πi: Ln(zw) = Ln z + Ln w + 2πik, with k = 0, 1 or −1.
• ln(zⁿ) = n ln z holds as a set equality for integer n only in the sense that n ln z gives some, but not all, values of ln(zⁿ). For example Ln((−1)²) = Ln 1 = 0, while 2 Ln(−1) = 2πi.

Example for the failure of the product rule with principal values: Ln((−1)(−1)) = Ln 1 = 0, but Ln(−1) + Ln(−1) = iπ + iπ = 2πi. These differ by 2πi, which is a valid value of the multi-valued logarithm.

How are complex powers defined?

For z ≠ 0 and any complex w: z^w = e^{w ln z}. Because ln z is multi-valued, z^w has in general infinitely many values: e^{w(Ln z + 2kπi)} = e^{w Ln z} · e^2kπiw. The principal value is e^{w Ln z}.

• For integer w = n: e^2kπin = 1, so there is only one value (the usual power).
• For rational w = p/q (in lowest terms): there are q distinct values, the qth roots of z^p.
• For irrational or complex w: infinitely many values.

Worked examples

Example 1: logarithms of special numbers

ln(−1) = ln 1 + i(π + 2kπ) = (2k + 1)πi. So the principal value is Ln(−1) = πi. ln i = ln 1 + i(π/2 + 2kπ) = i(π/2 + 2kπ). ln(−i) = i(−π/2 + 2kπ). ln(1 + i) = ln √2 + i(π/4 + 2kπ) = 0.3466 + 0.7854i + 2kπi.

Example 2: iⁱ

iⁱ = e^{i ln i} = e^{i · i(π/2 + 2kπ)} = e^{−π/2 − 2kπ}. The principal value is e^−π/2 ≈ 0.2079. All values are real: 0.2079, 0.000388 (k = 1), 111.3 (k = −1), …

Example 3: (−1)^1/3

(−1)^1/3 = e^{(1/3)(iπ + 2kπi)} = e^{iπ(2k + 1)/3}, with the values e^iπ/3 = 1/2 + (√3/2)i, e^iπ = −1 and e^i5π/3 = 1/2 − (√3/2)i. These are the cube roots of −1, while the real cube root −1 is just one of them.

Example 4: 2ⁱ

2ⁱ = e^{i ln 2} = cos(ln 2) + i sin(ln 2) = cos 0.6931 + i sin 0.6931 = 0.7692 + 0.6390i. The modulus is 1, since the exponent is purely imaginary and the base is real positive.

Example 5: (1 + i)^{2 − i}

Ln(1 + i) = 0.3466 + 0.7854i. Multiply by (2 − i): (0.3466)(2) + (0.7854)(1) + i[(0.7854)(2) − (0.3466)(1)] = (0.6931 + 0.7854) + i(1.5708 − 0.3466) = 1.4785 + 1.2242i. Then e^1.4785 = 4.386 and the angle 1.2242 rad gives cos 1.2242 = 0.3395, sin 1.2242 = 0.9406. The principal value is 4.386(0.3395 + 0.9406i) = 1.489 + 4.125i.

Example 6: solving e^z = 1 + i

z = ln(1 + i) = 0.3466 + i(π/4 + 2kπ). For k = 0: z = 0.3466 + 0.7854i. Check: e^0.3466 = 1.4142 = √2, and the angle 0.7854 = 45° gives √2(cos 45° + i sin 45°) = 1 + i ✓.

Example 7: a real-looking identity that fails

(e^2πi)ⁱ = 1ⁱ = e^{i ln 1} = e^i(2kπi) = e^−2kπ, which includes the values 1, e^−2π ≈ 0.00187, e^2π ≈ 535.5. But e^{2πi · i} = e^−2π is only one of those values. So the rule (a^b)^c = a^bc must be used with care.

Table of logarithms

Where does the idea come from?

The logarithm of a negative number was the subject of a famous dispute between Leibniz and Johann Bernoulli in 1712–1713: Bernoulli held that ln(−x) = ln x, while Leibniz argued that logarithms of negative numbers are imaginary. Euler settled the question in 1749, showing that the logarithm has infinitely many values, all differing by multiples of 2πi, as a consequence of his formula e^iθ = cos θ + i sin θ.

Where is it used today?

The complex logarithm is used in signal processing (the cepstrum and phase unwrapping), in the solution of linear differential equations and in complex analysis, where the Riemann surface of the logarithm is the standard example of a multi-valued function made single-valued on a surface. In engineering, the logarithm of a complex gain gives the decibel magnitude and the phase: ln(Ae^iφ) = ln A + iφ.

Which mistakes are common?

Treating ln z as single-valued

Specify the principal value Ln z, or give the general expression with 2kπi.

Applying real rules blindly

ln(z²) is not always 2 ln z, and (a^b)^c is not always a^bc for complex numbers.

Using the wrong argument for negative numbers

For the principal value, Arg(−5) = π, not −π.

Thinking iⁱ is imaginary

iⁱ = e^−π/2 is a real number.