  Determining the doping level in high-temperature cuprate superconductors is
crucial for understanding the origin of superconductivity in these materials
and for unlocking their full potential. However, accurately determining the
doping level remains a significant challenge due to a complex interplay of
factors and limitations in various measurement techniques. In particular, in
Bi$_{2}$Sr$_2$CuO$_{6+\delta}$ and Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$, where
the mobile carriers are introduced by non-stoichiometric oxygen $\delta$, the
determination has been extremely problematic. Here, we study the doping
dependence of the electronic structure of these materials in angle-resolved
photoemission and find that both the doping level, $p$, and the superconducting
transition temeprature, $T_c$ can be precisely determined from the binding
energy of the Bi $5d$ core-levels. The measurements can be performed at room
temperature, enabling the determination of $p$ and $T_c$ without cooling the
samples. This should be very helpful for further studies of these materials.
