  In this paper, we compare the scalar field dynamics in axion-like and power
law potentials for both positive and negative values of the exponents. We find
that, for positive exponents, both the potentials exhibit similar scalar field
dynamics and it can be difficult to distinguish them at least at the background
level. Even though the potentials are oscillatory in nature for positive
exponents scaling solutions can be achieved for larger values of the exponent
for which the dynamics can be different during early times. Because of the
presence of this scaling nature there is a turnaround in the values of the
scalar field equation of state as we increase the values of the exponent in
both the potentials. This indicates the deviation from the oscillatory
behaviour for the larger values of the exponent. For negative values of the
exponent, the dynamics of the scalar field is distinguishable and axion-like
potential can give rise to cosmologically viable tracker solutions unlike the
power law potentials. For negative values of the exponent, axion-like potential
can behave like a cosmological constant around its minima and the dark energy
scale can be related to the potential scale. Due to the cosmological constant
like behavior of the axion-like potential for negative exponent around its
minima the late time dynamics can be similar to $\Lambda$CDM and we get similar
observational constraint on the parameters for both $\Lambda$CDM and axion-like
potential with negative exponent. So, while for positive exponents we may not
distinguish the two potentials for negative exponents the dynamics of the
scalar field is distinguishable.
