parameters $$\begin{aligned}
 n_{s} &= 1 - 6 \epsilon_{\text{V}} + 2 \eta_{\text{V}}  \, , \label{ns} \\
r &= 16 \epsilon_{\text{V}} \label{r} \, , \\
\frac{d n_{s}}{d \ln k} &= 16 \epsilon_{\text{V}} \eta_{\text{V}} -24 \epsilon_{\text{V}}^2 -2 \xi_{\text{V}}^{2} \label{runns} \, , \\
A_{s} &= \frac{32}{75 \Mp^4} \frac{V}{\epsilon_{\text{V}}} \label{As} \, . \end{aligned}$$

Thus, if one solves the evolution of $f(R)$, then one can obtain the model predictions for the inflationary evolution. In order to obtain the numerical analysis for these predictions, one needs to solve the field equation obtained from Eq.(\[ESR2lnR\]) $$\ddot{\phi} + 3H \dot{\phi} + V_{,\phi} = 0 \label{phiEq} \, ,$$ where dot denotes the derivative with respect to the time, $t$. If one adopts the normalized field definition $\bMp \bphi = \phi$ and changes variable from $t$ to the e-folding number $N = \ln a$, then the above field equation becomes $$\bphi'' +  \left( 6 - \bphi^{'2} \right) \frac{\bphi'}{2} + \left( 6 - \bphi^{'2} \right) \frac{V_{,\bphi}}{2V} = 0 \label{bphiEq} \, .$$ The evolution of the given $f(R)$ as a function of $N$ is obtained when one inserts Eqs.(\[bphi\])-(\[V\]) into the Eq.(\[bphiEq\]). We investigate the specific model of $f(R)$ in the following subsection.

$R + \alpha \frac{R^2}{M^2} + \beta \frac{R^2}{M^2} \ln \frac{R}{M^2}$ model
