The article discusses certain equalities regarding powers and logarithms for positive real numbers, which do not hold true for complex numbers, regardless of how complex powers and logarithms are defined as single-valued functions.
- For all , holds true.
- For all and , also holds true, which can be proven using de Moivre’s theorem.
- However, for all and , does not hold true.
The selection of the principal value for multi-valued functions is the key point.