  We stude the \textit{holographic} QCD model which contains a quadratic term $
-\sigma z^2$ and a logarithmic term $-c_0\log[(z_{IR}-z)/z_{IR}]$ with an
explicit infrared cut-off $z_{IR}$ in the deformed ${\rm AdS}_5$ warp factor.
We investigate the heavy quark potential for three cases, i.e, with only
quadratic correction, with both quadratic and logarithmic corrections and with
only logarithmic correction. We solve the dilaton field and dilation potential
from the Einstein equation, and investigate the corresponding beta function in
the G{\"u}rsoy -Kiritsis-Nitti (GKN) framework. Our studies show that in the
case with only quadratic correction, a negative $\sigma$ or the
Andreev-Zakharov model is favored to fit the heavy quark potential and to
produce the QCD beta-function at 2-loop level, however, the dilaton potential
is unbounded in infrared regime. One interesting observing for the case of
positive $\sigma$, or the soft-wall ${\rm AdS}_5$ model is that the
corresponding beta-function exists an infrared fixed point. In the case with
only logarithmic correction, the heavy quark Cornell potential can be fitted
very well, the corresponding beta-function agrees with the QCD beta-function at
2-loop level reasonably well, and the dilaton potential is bounded from below
in infrared. At the end, we propose a more compact model which has only
logarithmic correction in the deformed warp factor and has less free
parameters.
