A family of curves with monotone curvature called the log-aesthetic curves (LAC) has been investigated in the field of industrial shape design. LAC has a radius of curvature in proportional to the power of a linear function of an arc-length parameter. In the present article we show that LAC can be naturally formulated in the similarity geometry and the Riccati equations satisfied by the similarity curvatures of LAC can be derived. Moreover, we clarify that certain generalizations of LAC (GLAC) can also be described in a uniform way.