• We can extend the definition of logarithm to nonzero complex numbers z if
we accept

that it has to be a multivalued function. We called this multivalued function

• Next we defined a^{b} for any a, b ∈C, with a
≠ 0:

This is a set, meaning it could potentially have more than one value!

• This seems a little crazy at first, but it’s not so bad.
For example we saw that this

definition of a^{b} as a set agrees with our intuition when b is a
rational number: if b = p/q

in lowest terms, then a^{b} will consist of q different values (e.g., 36^{1/2}
= {−6, +6}).

• But this definition also has interesting consequences
when b is not a rational number: if

b is irrational, then a^{b} will have infinitely many values! For
instance, if = 1.4142...

is the positive square root of 2, then

and these numbers are all different.

• I left it as an exercise to show that

which is a little unexpected: it says that all values of
are positive real numbers!

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