  The cosmological evolution within the framework of exponential $F(R)$ gravity
is analysed by assuming two forms for dark matter: (a) a standard dust-like
fluid and (b) an axion scalar field. As shown in previous literature, an
axion-like field oscillates during the cosmological evolution but can play the
role of dark matter when approaching the minimum of its potential. Both
scenarios are confronted with recent observational data including the Pantheon
Type Ia supernovae, Hubble parameter estimations (Cosmic Chronometers), Baryon
Acoustic Oscillations and Cosmic Microwave Background distances. The models
show great possibilities in describing these observations when compared with
the $\Lambda$CDM model, supporting the viability of exponential $F(R)$ gravity.
The differences between both descriptions of dark matter is analysed.
