Precalculus Examples

$x = \frac{e^{y}}{1 + 2 e^{y}}$

Step 1

Rewrite the equation as $\frac{e^{y}}{1 + 2 e^{y}} = x$ .

$\frac{e^{y}}{1 + 2 e^{y}} = x$

Step 2

Multiply both sides by $1 + 2 e^{y}$ .

$\frac{e^{y}}{1 + 2 e^{y}} (1 + 2 e^{y}) = x (1 + 2 e^{y})$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $1 + 2 e^{y}$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{e^{y}}{1 + 2 e^{y}} (1 + 2 e^{y}) = x (1 + 2 e^{y})$

Step 3.1.1.2

Rewrite the expression.

$e^{y} = x (1 + 2 e^{y})$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $x (1 + 2 e^{y})$ .

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Step 3.2.1.1

Apply the distributive property.

$e^{y} = x \cdot 1 + x (2 e^{y})$

Step 3.2.1.2

Simplify the expression.

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Step 3.2.1.2.1

Multiply $x$ by $1$ .

$e^{y} = x + x (2 e^{y})$

Step 3.2.1.2.2

Rewrite using the commutative property of multiplication.

$e^{y} = x + 2 x e^{y}$

Step 4

Solve for $y$ .

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Step 4.1

Subtract $2 x e^{y}$ from both sides of the equation.

$e^{y} - 2 x e^{y} = x$

Step 4.2

Factor $e^{y}$ out of $e^{y} - 2 x e^{y}$ .

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Step 4.2.1

Multiply by $1$ .

$e^{y} \cdot 1 - 2 x e^{y} = x$

Step 4.2.2

Factor $e^{y}$ out of $- 2 x e^{y}$ .

$e^{y} \cdot 1 + e^{y} (- 2 x) = x$

Step 4.2.3

Factor $e^{y}$ out of $e^{y} \cdot 1 + e^{y} (- 2 x)$ .

$e^{y} (1 - 2 x) = x$

Step 4.3

Divide each term in $e^{y} (1 - 2 x) = x$ by $1 - 2 x$ and simplify.

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Step 4.3.1

Divide each term in $e^{y} (1 - 2 x) = x$ by $1 - 2 x$ .

$\frac{e^{y} (1 - 2 x)}{1 - 2 x} = \frac{x}{1 - 2 x}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $1 - 2 x$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{e^{y} (1 - 2 x)}{1 - 2 x} = \frac{x}{1 - 2 x}$

Step 4.3.2.1.2

Divide $e^{y}$ by $1$ .

$e^{y} = \frac{x}{1 - 2 x}$

Step 4.4

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

$\ln (e^{y}) = \ln (\frac{x}{1 - 2 x})$

Step 4.5

Expand the left side.

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Step 4.5.1

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

$y \ln (e) = \ln (\frac{x}{1 - 2 x})$

Step 4.5.2

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

$y \cdot 1 = \ln (\frac{x}{1 - 2 x})$

Step 4.5.3

Multiply $y$ by $1$ .

$y = \ln (\frac{x}{1 - 2 x})$