Precalculus Examples

$\frac{500}{100 - e^{\frac{x}{2}}} = 20$

Step 1

Multiply both sides by $100 - e^{\frac{x}{2}}$ .

$\frac{500}{100 - e^{\frac{x}{2}}} (100 - e^{\frac{x}{2}}) = 20 (100 - e^{\frac{x}{2}})$

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 $100 - e^{\frac{x}{2}}$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{500}{100 - e^{\frac{x}{2}}} (100 - e^{\frac{x}{2}}) = 20 (100 - e^{\frac{x}{2}})$

Step 2.1.1.2

Rewrite the expression.

$500 = 20 (100 - e^{\frac{x}{2}})$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $20 (100 - e^{\frac{x}{2}})$ .

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Step 2.2.1.1

Apply the distributive property.

$500 = 20 \cdot 100 + 20 (- e^{\frac{x}{2}})$

Step 2.2.1.2

Multiply.

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Step 2.2.1.2.1

Multiply $20$ by $100$ .

$500 = 2000 + 20 (- e^{\frac{x}{2}})$

Step 2.2.1.2.2

Multiply $- 1$ by $20$ .

$500 = 2000 - 20 e^{\frac{x}{2}}$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $2000 - 20 e^{\frac{x}{2}} = 500$ .

$2000 - 20 e^{\frac{x}{2}} = 500$

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 $2000$ from both sides of the equation.

$- 20 e^{\frac{x}{2}} = 500 - 2000$

Step 3.2.2

Subtract $2000$ from $500$ .

$- 20 e^{\frac{x}{2}} = - 1500$

Step 3.3

Divide each term in $- 20 e^{\frac{x}{2}} = - 1500$ by $- 20$ and simplify.

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Step 3.3.1

Divide each term in $- 20 e^{\frac{x}{2}} = - 1500$ by $- 20$ .

$\frac{- 20 e^{\frac{x}{2}}}{- 20} = \frac{- 1500}{- 20}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 20$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 20 e^{\frac{x}{2}}}{- 20} = \frac{- 1500}{- 20}$

Step 3.3.2.1.2

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

$e^{\frac{x}{2}} = \frac{- 1500}{- 20}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $- 1500$ by $- 20$ .

$e^{\frac{x}{2}} = 75$

Step 3.4

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

$\ln (e^{\frac{x}{2}}) = \ln (75)$

Step 3.5

Expand the left side.

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Step 3.5.1

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

$\frac{x}{2} \ln (e) = \ln (75)$

Step 3.5.2

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

$\frac{x}{2} \cdot 1 = \ln (75)$

Step 3.5.3

Multiply $\frac{x}{2}$ by $1$ .

$\frac{x}{2} = \ln (75)$

Step 3.6

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \ln (75)$

Step 3.7

Simplify the left side.

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Step 3.7.1

Cancel the common factor of $2$ .

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Step 3.7.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \ln (75)$

Step 3.7.1.2

Rewrite the expression.

$x = 2 \ln (75)$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 2 \ln (75)$

Decimal Form:

$x = 8.63497622 \dots$