$P(\hat{Y})=\frac{e^{\beta_{0}+\beta_{1}X}}{1+e^{\beta_{0}+\beta_{1}X}}$ $\Rightarrow P(\hat{Y})+P(\hat{Y})\times[e^{\beta_{0}+\beta_{1}X}]=e^{\beta_{0}+\beta_{1}X}$ $\Rightarrow P(\hat{Y})=[1-P(\hat{Y})]\times [e^{\beta_{0}+\beta_{1}X}]$ $\Rightarrow \frac{P(\hat{Y})}{1-P(\hat{Y})}=e^{\beta_{0}+\beta_{1}X}$ The left hand side is called the odds. $\Rightarrow\log{\left(\frac{P(\hat{Y})}{1-P(\hat{Y})}\right)}=\beta_{0}+\beta_{1}X$ The left hand side is called the log-odds or logit. This models the log-odds as a linear function of $X$. The coefficient $\beta_{1}$ means: a one-unit increase in $X$ is associated with an increase in the log-odds by $\beta_{1}$ units.