The exponential function, logarithms or trigonometric functions are functions whereas a complex variable is an element of the complex field.

Each one of the functions can be defined for a complex variable.

Louis Legendre Pennisi has written:

'Elements of complex variable' -- subject(s): Functions of complex variables

In algebra, a variable is usually symbolized by the letter x. For more complex problems, the letters y and z are often used as well.

Yes and it is z=x+iy

Maurice Heins has written:

'Complex function theory' -- subject(s): Functions of complex variables

'Selected topics in the classical theoryof functions of a complex variable' -- subject(s): Functions of complex variables