10. The point \( (2,7) \) lies on the graph of \( y=e^{k x} \). Find the value of \( k \).

To find the value of \( k \), we start with the equation of the curve, \( y = e^{k x} \), and the point \( (2, 7) \). Substituting the coordinates of the point into the equation gives us: \[ 7 = e^{k \cdot 2} \] This simplifies to: \[ e^{2k} = 7 \] To solve for \( 2k \), we apply the natural logarithm to both sides: \[ \ln(e^{2k}) = \ln(7) \] This simplifies to: \[ 2k = \ln(7) \] Next, we divide both sides by 2 to solve for \( k \): \[ k = \frac{\ln(7)}{2} \] Thus, the value of \( k \) is \[ \boxed{\frac{\ln(7)}{2}}. \]