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The values of a and b are not restricted to real numbers; complex numbers are allowed (they are never rational when they have an imaginary part not equal to 0, even if both the real and imaginary parts are rational).
In general, is multivalued, where "log" stands for the complex logarithm. This accounts for the phrase "any value of" in the theorem's statement.
An equivalent formulation of the theorem is the following: if ? and ? are nonzero algebraic numbers, and we take any non-zero logarithm of ?, then is either rational or transcendental. This may be expressed as saying that if , are linearly independent over the rationals, then they are linearly independent over the algebraic numbers. The generalisation of this statement to more general linear forms in logarithms of several algebraic numbers is in the domain of transcendental number theory.
If the restriction that a and b be algebraic is removed, the statement does not remain true in general. For example,
Here, a is , which (as proven by the theorem itself) is transcendental rather than algebraic. Similarly, if and , which is transcendental, then is algebraic. A characterization of the values for a and b, which yield a transcendental ab, is not known.