Please help me. How I should change ln(t) to "e to the something"?

Question

This problem is about parametric function and I have to eliminate parametric function.

x=lnt y=root t

I can see that y=e^x/2

How does the x=lnt change to "e to the something"?

Answer 1

To convert from parametric to rectangular solve one of the parametric equations for t, then substitute into the other.

Solve y = √t for t to get t = y².

Substitute that value for t into the other.

x = ln(y²)

x=2ln(y)

ln(y) = x/2

Now convert to exponential form by raising the base of the logarithm (e) to the logarithm (x/2) to generate the antilogarithm (y).

y = e^x/2

Now try it the other way:

If x = ln(t), then t = e^x [raise the base of the logarithm (e) to the logarithm (x) to get the antilogarithm (t)]

Now substitute that value for t (in terms of x) into the parametric for y:

y =√e^x=(e^x)^1/2 = e^x/2