I guess when you say you want a cartesian equation of the curve, you would have to express them in cartesian coordinates of x and y. Thus, you must get rid of the variable t. The solution is as follows:
x = t⁴
y = ln(t)
e^y = e^(lnt)
e^y = t
Substuting this to the first equation,
x = (e^y)⁴
x = e^4y
