Calculus Examples

$x = \ln (t)$ $y = \sqrt{t}$

Step 1

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

$x = \ln (t)$

Step 2

Rewrite the equation as $\ln (t) = x$ .

$\ln (t) = x$

Step 3

To solve for $t$ , rewrite the equation using properties of logarithms.

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

Step 4

Rewrite $\ln (t) = x$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{x} = t$

Step 5

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

$t = e^{x}$

Step 6

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

$y = \sqrt{e^{x}}$

Step 7

Remove parentheses.

$y = \sqrt{e^{x}}$