Question

the set of all points $(e^t, t)$, where $t$ is a real number, is the graph of $y = $

Explanation:

Step1: Define variables from the parametric form

Given the parametric equations \( x = e^t \) and \( y = t \) (since the points are \( (e^t, t) \)).

Step2: Solve for \( t \) from the \( y \)-equation

From \( y = t \), we can directly see that \( t = y \).

Step3: Substitute \( t \) into the \( x \)-equation

Substitute \( t = y \) into \( x = e^t \). So we get \( x = e^y \).

Step4: Solve for \( y \) in terms of \( x \)

To express \( y \) in terms of \( x \), we take the natural logarithm of both sides of \( x = e^y \). By the definition of logarithms, if \( x = e^y \), then \( y=\ln(x) \) (since the natural logarithm is the inverse of the exponential function \( e^x \)).

Answer:

\(\ln(x)\)