  We define a point-free construction of real exponentiation and logarithms,
i.e.\ we construct the maps $\exp\colon (0, \infty)\times \mathbb{R}
\rightarrow \!(0,\infty),\, (x, \zeta) \mapsto x^\zeta$ and $\log\colon
(1,\infty)\times (0, \infty) \rightarrow\mathbb{R},\, (b, y) \mapsto
\log_b(y)$, and we develop familiar algebraic rules for them. The point-free
approach is constructive, and defines the points of a space as models of a
geometric theory, rather than as elements of a set - in particular, this allows
geometric constructions to be applied to points living in toposes other than
Set. Our geometric development includes new lifting and gluing techniques in
point-free topology, which highlight how properties of $\mathbb{Q}$ determine
properties of real exponentiation.
  This work is motivated by our broader research programme of developing a
version of adelic geometry via topos theory. In particular, we wish to
construct the classifying topos of places of $\mathbb{Q}$, which will provide a
geometric perspective into the subtle relationship between $\mathbb{R}$ and
$\mathbb{Q}_p$, a question of longstanding number-theoretic interest.
