We introduce a new logotropic model based on a complex scalar field with a logarithmic potential that unifies dark matter and dark energy. The scalar field satisfies a nonlinear wave equation generalizing the Klein-Gordon equation in the relativistic regime and the Schrödinger equation in the nonrelativistic regime. This model has an intrinsically quantum nature and returns the $\mathrm{\Lambda }$ cold dark matter (CDM) model in the classical limit $\hslash \to 0$. It involves a new fundamental constant of physics $A/c^2=2.10\times {10}^{-26}\mathrm{g}{\mathrm{m}}^{-3}$ responsible for the late accelerating expansion of the Universe and superseding the Einstein cosmological constant $\mathrm{\Lambda }$. The logotropic model is almost indistinguishable from the $\mathrm{\Lambda }\mathrm{CDM}$ model at large (cosmological) scales but solves the CDM crisis at small (galactic) scales. It also solves the problems of the fuzzy dark matter model. Indeed, it leads to cored dark matter halos with a universal surface density ${\mathrm{\Sigma }}_0^{\mathrm{th}}=5.85(A/4\pi G{)}^{1/2}=133M_{\odot }/{\mathrm{pc}}^2$. This universal surface density is predicted from the logotropic model without adjustable parameter and turns out to be close to the observed value ${\mathrm{\Sigma }}_0^{\mathrm{obs}}=141_{-52}^{+83}{M}_{\odot }/{\mathrm{pc}}^2$. We also argue that the quantities ${\mathrm{\Omega }}_{\mathrm{dm},0}$ and ${\mathrm{\Omega }}_{\mathrm{de},0}$, which are usually interpreted as the present proportions of dark matter and dark energy in the $\mathrm{\Lambda }\mathrm{CDM}$ model, are equal to ${\mathrm{\Omega }}_{\mathrm{dm},0}^{\mathrm{th}}=\frac{1}{1+e}(1-{\mathrm{\Omega }}_{\mathrm{b},0})=0.2559$ and ${\mathrm{\Omega }}_{\mathrm{de},0}^{\mathrm{th}}=\frac{e}{1+e}(1-{\mathrm{\Omega }}_{\mathrm{b},0})=0.6955$ in very good agreement with the measured values ${\mathrm{\Omega }}_{\mathrm{dm},0}^{\mathrm{obs}}=0.2589$ and ${\mathrm{\Omega }}_{\mathrm{de},0}^{\mathrm{obs}}=0.6911$ (their empirical ratio 2.669 is close to the Euler number $e=2.71828\dots$). We point out, however, important difficulties with the logotropic model, similar to those encountered by the generalized Chaplygin gas model. These problems are related to the difficulty of forming large-scale structures due to an increasing speed of sound as the Universe expands. We discuss potential solutions to these problems, stressing in particular the importance to perform a nonlinear study of structure formation.