  We study the properties of Q-balls dominated by the thermal logarithmic
potential analytically instead of estimating the characters with only some
specific values of model variables numerically. In particular the analytical
expressions for radius and energy of this kind of Q-ball are obtained.
According to these explicit expressions we demonstrate strictly that the large
Q-balls enlarge and the small ones become smaller in the background with lower
temperature. The energy per unit charge will not be divergent if the charge is
enormous. We find that the lower temperature will lead the energy per unit
charge of Q-ball smaller. We also prove rigorously the necessary conditions
that the model parameters should satisfy to keep the stability of the Q-balls.
When one of model parameters of Q-balls $K$ is positive, the Q-balls will not
form or survive unless the temperature is high enough. In the case of negative
$K$, the Q-balls are stable no matter the temperature is high or low.
