Am I missing a step in the elimination of parameter $t$?

Question

I am working on eliminating parameter $t$ in these parametric equations:

1. $x(t) = e^{2t}$

2. $y(t) = e^t + 3$

I know when eliminating parameter $t$, you have to solve for $t$ in one equation and then plug the new equation into y. That's what I did for x and ended up with

$$t=\frac{ln(x)}{2}$$

Next, I just plugged this into $y$ and got

$$y=e^{\frac{ln(x)}{2}}+3$$

Unfortunately, none of the answers provided by the quiz match my answer, the only options are:

1. $y=e^{2x}+3$

2. $y=\sqrt{x-3}$

3. $y=x^2 +3$

4. $y=\sqrt{x}+3$

Did I accidentally miss a step?

Answer 1

You have

$$e^\frac{\ln(x)}{2}=(e^{\ln(x)})^\frac{1}{2}=x^\frac{1}{2}=\sqrt{x}$$

Thus, you have

$$e^\frac{\ln(x)}{2}+3=\sqrt{x}+3$$, i.e. number 4 is right.

Answer 2

Notice that $x(t) = e^{2t} = (e^t)^2 = (y(t)-3)^2$.