Writing parametric curve in Cartesian form

Question

$$x=4 \ln(t), \qquad y=2t$$

Write it in Cartesian form, giving $y$ explicitly in terms of $x$.

Answer 1

What you need to do is to eliminate $t$ from both expressions (or equivalently, write $t=f(x)$ or $t=f(y)$ and substitute into the other expression).

In this case, $t=y/2$, so we have $$x=4\ln(y/2),\mbox{ or }y=2\exp((1/4) x),$$ or $t=\exp((1/4) x)$, so $$y=2\exp((1/4) x).$$