The John inclusion for log-concave functions

John’s inclusion states that a convex body in ${\mathbb{R}}^d$ can be covered by the $d$-dilation of its maximal volume ellipsoid. We obtain a certain John-type inclusion for log-concave functions. As a byproduct of our approach, we establish the following asymptotically tight inequality: \ \noindent For any log-concave function $f$ with finite, positive integral, there exist a positive definite matrix $A$, a point $a\in {\mathbb{R}}^d$, and a positive constant $\alpha$ such that $${\chi }_{{\mathbf{B}}^d}(x)\le \alpha f\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}(A(x-a))\le \sqrt{d+1}\cdot e^{-\frac{\mid x\mid }{d+2}+(d+1)},$$ where ${\chi }_{{\mathbf{B}}^d}$ is the indicator function of the unit ball ${\mathbf{B}}^d$.