Precalculus Examples

$\frac{50}{4 + e^{2 x}} = 11$

Step 1

Multiply both sides by $4 + e^{2 x}$ .

$\frac{50}{4 + e^{2 x}} (4 + e^{2 x}) = 11 (4 + e^{2 x})$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $4 + e^{2 x}$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{50}{4 + e^{2 x}} (4 + e^{2 x}) = 11 (4 + e^{2 x})$

Step 2.1.1.2

Rewrite the expression.

$50 = 11 (4 + e^{2 x})$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $11 (4 + e^{2 x})$ .

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Step 2.2.1.1

Apply the distributive property.

$50 = 11 \cdot 4 + 11 e^{2 x}$

Step 2.2.1.2

Multiply $11$ by $4$ .

$50 = 44 + 11 e^{2 x}$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $44 + 11 e^{2 x} = 50$ .

$44 + 11 e^{2 x} = 50$

Step 3.2

Move all terms not containing $x$ to the right side of the equation.

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Step 3.2.1

Subtract $44$ from both sides of the equation.

$11 e^{2 x} = 50 - 44$

Step 3.2.2

Subtract $44$ from $50$ .

$11 e^{2 x} = 6$

Step 3.3

Divide each term in $11 e^{2 x} = 6$ by $11$ and simplify.

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Step 3.3.1

Divide each term in $11 e^{2 x} = 6$ by $11$ .

$\frac{11 e^{2 x}}{11} = \frac{6}{11}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $11$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{11 e^{2 x}}{11} = \frac{6}{11}$

Step 3.3.2.1.2

Divide $e^{2 x}$ by $1$ .

$e^{2 x} = \frac{6}{11}$

Step 3.4

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

$\ln (e^{2 x}) = \ln (\frac{6}{11})$

Step 3.5

Expand the left side.

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Step 3.5.1

Expand $\ln (e^{2 x})$ by moving $2 x$ outside the logarithm.

$2 x \ln (e) = \ln (\frac{6}{11})$

Step 3.5.2

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

$2 x \cdot 1 = \ln (\frac{6}{11})$

Step 3.5.3

Multiply $2$ by $1$ .

$2 x = \ln (\frac{6}{11})$

Step 3.6

Divide each term in $2 x = \ln (\frac{6}{11})$ by $2$ and simplify.

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Step 3.6.1

Divide each term in $2 x = \ln (\frac{6}{11})$ by $2$ .

$\frac{2 x}{2} = \frac{\ln (\frac{6}{11})}{2}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $2$ .

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Step 3.6.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\ln (\frac{6}{11})}{2}$

Step 3.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{6}{11})}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (\frac{6}{11})}{2}$

Decimal Form:

$x = - 0.30306790 \dots$