A complex logarithm extends the concept of logarithms to complex numbers, which are numbers that have both a real part and an imaginary part. For a complex number expressed as z = re^(iθ), where r is the magnitude and θ is the angle (or argument), the complex logarithm is defined as log(z) = log(r) + iθ. This allows us to find logarithmic values for numbers that are not purely real. The complex logarithm is multi-valued due to the periodic nature of the e^(iθ) function, meaning that there are infinitely many values for the logarithm of a complex number. This is often represented using a principal value, typically taken when θ is in the range of (-π, π]. Understanding complex logarithms is essential in fields like complex analysis and signal processing.