Constructing parametric equation for $x=3z\cos(\ln z)$

Question

I was trying to transform this $$x=3z\cos(\ln z)$$ in parametric form: $$x=x(t)$$ $$z=z(t).$$ To this end I made a substitution $\ln{z}=t$ and I got: $$x=3{e^t}\cos(t)$$ $$z=e^t$$ $$t\in (-\infty,\infty)$$ But the graphs of initial equation and the resultant one are completely different. Please show me my mistake.

Answer 1

Easiest way to parametrize this is by putting $z = t$, then $x = 3t \cos \ln t$, $t \in (-\infty, \, \infty )$.