Precalculus Examples

$78 = \frac{90}{1 + 271 e^{- 0.122 X}}$

Step 1

Rewrite the equation as $\frac{90}{1 + 271 e^{- 0.122 X}} = 78$ .

$\frac{90}{1 + 271 e^{- 0.122 X}} = 78$

Step 2

Multiply both sides by $1 + 271 e^{- 0.122 X}$ .

$\frac{90}{1 + 271 e^{- 0.122 X}} (1 + 271 e^{- 0.122 X}) = 78 (1 + 271 e^{- 0.122 X})$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Simplify the left side.

Tap for more steps...

Step 3.1.1

Cancel the common factor of $1 + 271 e^{- 0.122 X}$ .

Tap for more steps...

Step 3.1.1.1

Cancel the common factor.

$\frac{90}{1 + 271 e^{- 0.122 X}} (1 + 271 e^{- 0.122 X}) = 78 (1 + 271 e^{- 0.122 X})$

Step 3.1.1.2

Rewrite the expression.

$90 = 78 (1 + 271 e^{- 0.122 X})$

Step 3.2

Simplify the right side.

Tap for more steps...

Step 3.2.1

Simplify $78 (1 + 271 e^{- 0.122 X})$ .

Tap for more steps...

Step 3.2.1.1

Apply the distributive property.

$90 = 78 \cdot 1 + 78 (271 e^{- 0.122 X})$

Step 3.2.1.2

Multiply.

Tap for more steps...

Step 3.2.1.2.1

Multiply $78$ by $1$ .

$90 = 78 + 78 (271 e^{- 0.122 X})$

Step 3.2.1.2.2

Multiply $271$ by $78$ .

$90 = 78 + 21138 e^{- 0.122 X}$

Step 4

Solve for $X$ .

Tap for more steps...

Step 4.1

Rewrite the equation as $78 + 21138 e^{- 0.122 X} = 90$ .

$78 + 21138 e^{- 0.122 X} = 90$

Step 4.2

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

Tap for more steps...

Step 4.2.1

Subtract $78$ from both sides of the equation.

$21138 e^{- 0.122 X} = 90 - 78$

Step 4.2.2

Subtract $78$ from $90$ .

$21138 e^{- 0.122 X} = 12$

Step 4.3

Divide each term in $21138 e^{- 0.122 X} = 12$ by $21138$ and simplify.

Tap for more steps...

Step 4.3.1

Divide each term in $21138 e^{- 0.122 X} = 12$ by $21138$ .

$\frac{21138 e^{- 0.122 X}}{21138} = \frac{12}{21138}$

Step 4.3.2

Simplify the left side.

Tap for more steps...

Step 4.3.2.1

Cancel the common factor of $21138$ .

Tap for more steps...

Step 4.3.2.1.1

Cancel the common factor.

$\frac{21138 e^{- 0.122 X}}{21138} = \frac{12}{21138}$

Step 4.3.2.1.2

Divide $e^{- 0.122 X}$ by $1$ .

$e^{- 0.122 X} = \frac{12}{21138}$

Step 4.3.3

Simplify the right side.

Tap for more steps...

Step 4.3.3.1

Cancel the common factor of $12$ and $21138$ .

Tap for more steps...

Step 4.3.3.1.1

Factor $6$ out of $12$ .

$e^{- 0.122 X} = \frac{6 (2)}{21138}$

Step 4.3.3.1.2

Cancel the common factors.

Tap for more steps...

Step 4.3.3.1.2.1

Factor $6$ out of $21138$ .

$e^{- 0.122 X} = \frac{6 \cdot 2}{6 \cdot 3523}$

Step 4.3.3.1.2.2

Cancel the common factor.

$e^{- 0.122 X} = \frac{6 \cdot 2}{6 \cdot 3523}$

Step 4.3.3.1.2.3

Rewrite the expression.

$e^{- 0.122 X} = \frac{2}{3523}$

Step 4.4

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

$\ln (e^{- 0.122 X}) = \ln (\frac{2}{3523})$

Step 4.5

Expand the left side.

Tap for more steps...

Step 4.5.1

Expand $\ln (e^{- 0.122 X})$ by moving $- 0.122 X$ outside the logarithm.

$- 0.122 X \ln (e) = \ln (\frac{2}{3523})$

Step 4.5.2

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

$- 0.122 X \cdot 1 = \ln (\frac{2}{3523})$

Step 4.5.3

Multiply $- 0.122$ by $1$ .

$- 0.122 X = \ln (\frac{2}{3523})$

Step 4.6

Divide each term in $- 0.122 X = \ln (\frac{2}{3523})$ by $- 0.122$ and simplify.

Tap for more steps...

Step 4.6.1

Divide each term in $- 0.122 X = \ln (\frac{2}{3523})$ by $- 0.122$ .

$\frac{- 0.122 X}{- 0.122} = \frac{\ln (\frac{2}{3523})}{- 0.122}$

Step 4.6.2

Simplify the left side.

Tap for more steps...

Step 4.6.2.1

Cancel the common factor of $- 0.122$ .

Tap for more steps...

Step 4.6.2.1.1

Cancel the common factor.

$\frac{- 0.122 X}{- 0.122} = \frac{\ln (\frac{2}{3523})}{- 0.122}$

Step 4.6.2.1.2

Divide $X$ by $1$ .

$X = \frac{\ln (\frac{2}{3523})}{- 0.122}$

Step 4.6.3

Simplify the right side.

Tap for more steps...

Step 4.6.3.1

Move the negative in front of the fraction.

$X = - \frac{\ln (\frac{2}{3523})}{0.122}$

Step 4.6.3.2

Replace $e$ with an approximation.

$X = - \frac{\log_{2.71828182} (\frac{2}{3523})}{0.122}$

Step 4.6.3.3

Divide $2$ by $3523$ .

$X = - \frac{\log_{2.71828182} (0.00056769)}{0.122}$

Step 4.6.3.4

Log base $2.71828182$ of $0.00056769$ is approximately $- 7.47392099$ .

$X = - \frac{- 7.47392099}{0.122}$

Step 4.6.3.5

Divide $- 7.47392099$ by $0.122$ .

$X = - - 61.26164752$

Step 4.6.3.6

Multiply $- 1$ by $- 61.26164752$ .

$X = 61.26164752$