The logarithmic function in base a is the inverse function of the exponential function in base a.
It is continuous.
The points (1, 0) and (a, 1) belong to the graph.
Increasing if a > 1.
Decreasing if a < 1.
The graph of the logarithmic function is symmetrical (with respect to the bisector of the 1st and 3rd quadrant) to the graph of the exponential function, as they are the inverse functions of each other.