Precalculus Examples

$x = e^{4 t}$ , $y = e^{- t}$

Step 1

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

$x = e^{4 t}$

Step 2

Rewrite the equation as $e^{4 t} = x$ .

$e^{4 t} = x$

Step 3

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

$\ln (e^{4 t}) = \ln (x)$

Step 4

Expand the left side.

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Step 4.1

Expand $\ln (e^{4 t})$ by moving $4 t$ outside the logarithm.

$4 t \ln (e) = \ln (x)$

Step 4.2

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

$4 t \cdot 1 = \ln (x)$

Step 4.3

Multiply $4$ by $1$ .

$4 t = \ln (x)$

Step 5

Divide each term in $4 t = \ln (x)$ by $4$ and simplify.

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Step 5.1

Divide each term in $4 t = \ln (x)$ by $4$ .

$\frac{4 t}{4} = \frac{\ln (x)}{4}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $4$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{4 t}{4} = \frac{\ln (x)}{4}$

Step 5.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (x)}{4}$

Step 6

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

$y = e^{- (\frac{\ln (x)}{4})}$

Step 7

Simplify $e^{- (\frac{\ln (x)}{4})}$ .

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Step 7.1

Rewrite $\frac{\ln (x)}{4}$ as $\frac{1}{4} \ln (x)$ .

$y = e^{- (\frac{1}{4} \ln (x))}$

Step 7.2

Simplify $\frac{1}{4} \ln (x)$ by moving $\frac{1}{4}$ inside the logarithm.

$y = e^{- \ln (x^{\frac{1}{4}})}$

Step 7.3

Simplify $- \ln (x^{\frac{1}{4}})$ by moving $- 1$ inside the logarithm.

$y = e^{\ln ({(x^{\frac{1}{4}})}^{- 1})}$

Step 7.4

Exponentiation and log are inverse functions.

$y = {(x^{\frac{1}{4}})}^{- 1}$

Step 7.5

Multiply the exponents in ${(x^{\frac{1}{4}})}^{- 1}$ .

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Step 7.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = x^{\frac{1}{4} \cdot - 1}$

Step 7.5.2

Combine $\frac{1}{4}$ and $- 1$ .

$y = x^{\frac{- 1}{4}}$

Step 7.5.3

Move the negative in front of the fraction.

$y = x^{- \frac{1}{4}}$

Step 7.6

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$y = \frac{1}{x^{\frac{1}{4}}}$