Precalculus Examples

$\frac{10}{1 + e^{- x}} = 2$

Step 1

Multiply both sides by $1 + e^{- x}$ .

$\frac{10}{1 + e^{- x}} (1 + e^{- x}) = 2 (1 + e^{- x})$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Simplify the left side.

Tap for more steps...

Step 2.1.1

Cancel the common factor of $1 + e^{- x}$ .

Tap for more steps...

Step 2.1.1.1

Cancel the common factor.

$\frac{10}{1 + e^{- x}} (1 + e^{- x}) = 2 (1 + e^{- x})$

Step 2.1.1.2

Rewrite the expression.

$10 = 2 (1 + e^{- x})$

Step 2.2

Simplify the right side.

Tap for more steps...

Step 2.2.1

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

Tap for more steps...

Step 2.2.1.1

Apply the distributive property.

$10 = 2 \cdot 1 + 2 e^{- x}$

Step 2.2.1.2

Multiply $2$ by $1$ .

$10 = 2 + 2 e^{- x}$

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $2 + 2 e^{- x} = 10$ .

$2 + 2 e^{- x} = 10$

Step 3.2

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

Tap for more steps...

Step 3.2.1

Subtract $2$ from both sides of the equation.

$2 e^{- x} = 10 - 2$

Step 3.2.2

Subtract $2$ from $10$ .

$2 e^{- x} = 8$

Step 3.3

Divide each term in $2 e^{- x} = 8$ by $2$ and simplify.

Tap for more steps...

Step 3.3.1

Divide each term in $2 e^{- x} = 8$ by $2$ .

$\frac{2 e^{- x}}{2} = \frac{8}{2}$

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.2.1.1

Cancel the common factor.

$\frac{2 e^{- x}}{2} = \frac{8}{2}$

Step 3.3.2.1.2

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

$e^{- x} = \frac{8}{2}$

Step 3.3.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.1

Divide $8$ by $2$ .

$e^{- x} = 4$

Step 3.4

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

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

Step 3.5

Expand the left side.

Tap for more steps...

Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

Multiply $- 1$ by $1$ .

$- x = \ln (4)$

Step 3.6

Divide each term in $- x = \ln (4)$ by $- 1$ and simplify.

Tap for more steps...

Step 3.6.1

Divide each term in $- x = \ln (4)$ by $- 1$ .

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

Step 3.6.2

Simplify the left side.

Tap for more steps...

Step 3.6.2.1

Dividing two negative values results in a positive value.

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

Step 3.6.2.2

Divide $x$ by $1$ .

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

Step 3.6.3

Simplify the right side.

Tap for more steps...

Step 3.6.3.1

Move the negative one from the denominator of $\frac{\ln (4)}{- 1}$ .

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

Step 3.6.3.2

Rewrite $- 1 \cdot \ln (4)$ as $- \ln (4)$ .

$x = - \ln (4)$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \ln (4)$

Decimal Form:

$x = - 1.38629436 \dots$