Mathematics Question from JEE Main 2023 · 10 April, Shift 2

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 $PA : PB = 1 : k$ and $y = y(x)$ is the solution of the differential equation $e^{\frac{dy}{dx}} = kx + \frac{k}{2}$, $y(0) = k$, then $4y(1) - 5 \log_e 3$ is equal to _____.