A particle moves along the \(x\)-axis with a velocity given by \(v(t) = v_0 e^{-kt}\), where \(v_0\) and \(k\) are positive constants. The particle is located at \(x = 0\) at time \(t = 0\), and as \(t \to \infty\) it approaches a maximum total displacement \(x_{\text{max}}\). In terms of \(k\), at what time \(t\) does the particle reach a position of \(\dfrac{1}{2}x_{\text{max}}\)?