A company produces a product using two inputs: labor $(L)$ and capital $(K).$ The price of labor $($wage rate$)$ is $$5$ per hour, and the price of capital $($rental rate$)$ is $$10$ per unit. The company's production function is given by $Q=ln(L^{0\mathrm{.}8}{K}^{0\mathrm{.}4}),$ where $Q$ is the quantity of output.

The company aims to produce $50$ units of the product. Determine the cost$-$minimizing combination of labor and capital the company should use. Detail your answer. Do not even think of using chatGPT.