Precalculus Examples

$x (t) = e^{t}$ , $y (t) = - e^{3 t}$

Step 1

Set up the parametric equation for $x (t)$ to solve the equation for $t$ .

$x = e^{t}$

Step 2

Rewrite the equation as $e^{t} = x$ .

$e^{t} = x$

Step 3

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{t}) = \ln (x)$

Step 4

Expand the left side.

Tap for more steps...

Step 4.1

Expand $\ln (e^{t})$ by moving $t$ outside the logarithm.

$t \ln (e) = \ln (x)$

Step 4.2

The natural logarithm of $e$ is $1$ .

$t \cdot 1 = \ln (x)$

Step 4.3

Multiply $t$ by $1$ .

$t = \ln (x)$

Step 5

Replace $t$ in the equation for $y$ to get the equation in terms of $x$ .

$y = - e^{3 (\ln (x))}$

Step 6

Simplify $- e^{3 (\ln (x))}$ .

Tap for more steps...

Step 6.1

Simplify $3 (\ln (x))$ by moving $3$ inside the logarithm.

$y = - e^{\ln (x^{3})}$

Step 6.2

Exponentiation and log are inverse functions.

$y = - x^{3}$