What is the codomain of the exponential function?

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Do we have

  1. $\exp:\mathbb R \rightarrow \mathbb R$; or
  2. $\exp:\mathbb R \rightarrow \mathbb R^+$?

Option 1 seems to be common but seems to have the problem that $\exp$ wouldn't be the inverse of $\ln:\mathbb R^+ \rightarrow \mathbb R $ ?

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The codomain may be any set that contains the positive real numbers. The range of course is the set of all positive reals, provided the domain is restricted to only the real numbers.