We present a new logarithmic Sobolev inequality adapted to a
log-concave measure on $\mathbb{R}$ between the exponential and the
Gaussian measure. More precisely, assume that $\Phi$ is a
symmetric convex function on $\mathbb{R}$ satisfying $(1+\varepsilon)\Phi(x)\leq
{x}\Phi'(x)\leq(2-\varepsilon)\Phi(x)$ for $x\geq 0$ large enough and with
$\varepsilon\in ]0,1/2]$. We prove that the probability measure on $\mathbb{R}$
$\mu_\Phi(dx)=e^{-\Phi(x)}/Z_\Phi dx$ satisfies a modified and
adapted logarithmic Sobolev inequality: there exist three
constants $A,B,C>0$ such that for all smooth functions $f>0$,
$$
\mathbf{Ent}_{\mu_\Phi}{\left(f^2\right)}\leq
A\int H_{\Phi}\left(\frac{f'}{f}\right)f^2d\mu_\Phi,
$$
with
$$
H_{\Phi}(x)=
\left\{
\begin{array}{l}
x^2 \text{ if }\left|x\right|< C,\\
\Phi^*\left(Bx\right) \text{ if }\left|x\right|\geq C,
\end{array}
\right.
$$
where $\Phi^*$ is the Legendre-Fenchel transform of $\Phi$.