\begin{array}{lcl}
     f[\theta(x_1,x_2,\dots,x_n)+(1-\theta)(y_1,y_2,\dots,y_n)]&=&f\left[\theta x_1+(1-\theta)y_1,\theta x_2+(1-\theta)y_2,\dots,\theta x_n+(1-\theta)y_n\right]\cr
          &=&\log[k_1(\theta x_1+(1-\theta)y_1)+k_2(\theta x_2+(1-\theta)y_2)+\dots+k_n(\theta x_n+(1-\theta)y_n)]\cr
           &=&\log(\theta(k_1x_1+k_2x_2+\dots+k_nx_n)+(1-\theta)(k_1y_1+k_2y_2+\dots+k_nx_n))\cr &\geq& \theta\log(k_1x_1+k_2x_2+\dots+k_nx_n)+(1-\theta)\log(k_1y_1+k_2y_2+\dots+k_ny_n)\cr &=&\theta f(x_1,x_2,\dots,x_n)+(1-\theta)f(y_1,y_2,\dots,y_n)

\end{array}