Trigonometry Examples

$y = x e^{a t}$

Step 1

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

$x e^{a t} = y$

Step 2

Divide each term in $x e^{a t} = y$ by $x$ and simplify.

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Step 2.1

Divide each term in $x e^{a t} = y$ by $x$ .

$\frac{x e^{a t}}{x} = \frac{y}{x}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $x$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{x e^{a t}}{x} = \frac{y}{x}$

Step 2.2.1.2

Divide $e^{a t}$ by $1$ .

$e^{a t} = \frac{y}{x}$

Step 3

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

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

Step 4

Expand the left side.

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Step 4.1

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

$a t \ln (e) = \ln (\frac{y}{x})$

Step 4.2

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

$a t \cdot 1 = \ln (\frac{y}{x})$

Step 4.3

Multiply $a$ by $1$ .

$a t = \ln (\frac{y}{x})$

Step 5

Divide each term in $a t = \ln (\frac{y}{x})$ by $t$ and simplify.

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Step 5.1

Divide each term in $a t = \ln (\frac{y}{x})$ by $t$ .

$\frac{a t}{t} = \frac{\ln (\frac{y}{x})}{t}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $t$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{a t}{t} = \frac{\ln (\frac{y}{x})}{t}$

Step 5.2.1.2

Divide $a$ by $1$ .

$a = \frac{\ln (\frac{y}{x})}{t}$