Question

consider the parametric curve defined by

$x = \ln(5t)$ and $y = 1 - t$.

eliminate the parameter to find the cartesian equation of the curve in the form $y = f(x)$.

$y =$

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Explanation:

Step1: Solve for \(t\) from \(x = \ln(5t)\)

By the property of logarithms, if \(x=\ln(5t)\), then \(e^{x}=5t\), and \(t = \frac{e^{x}}{5}\).

Step2: Substitute \(t\) into \(y = 1 - t\)

Substitute \(t=\frac{e^{x}}{5}\) into \(y = 1 - t\). We get \(y=1-\frac{e^{x}}{5}\).

Answer:

\(y = 1-\frac{e^{x}}{5}\)