Let the tangent at any point P on a curve passing through the points $(1,1)$ and $\left(\frac{1}{10}, 100\right)$, intersect positive $x$-axis and $y$-axis at the points $A$ and $B$ respectively. If PA. $P B=1$. $k$ and $y=y(x)$ is the solution of the differential equation $e^{\frac{d y}{d x}}=k x+\frac{k}{2}, y(0)=k$, then $4 y(1)-5 \log _e 3$ is equal to $\_\_\_\_$ .