The equation for a circle is a function in that it can be graphed and charted. One common equation is x^2 + y^2 = r^2.
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You can tell if an equation is a function if for any x value that you put into the function, you get only one y value. The equation you asked about is the equation of a line. It is a function.
Equation of the circle: (x-3)^2 +( y+13)^2 = 169
a function rule
The set of all values of x, for which the equation is true is the domain of the function defined by that equation.
A logarithmic equation would be any equation that includes the log function.