Calculus Examples

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

Step 1

To solve for $x$ , rewrite the equation using properties of logarithms.

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

Step 2

Rewrite $\ln (1 - x) = \frac{1}{4}$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{\frac{1}{4}} = 1 - x$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $1 - x = e^{\frac{1}{4}}$ .

$1 - x = e^{\frac{1}{4}}$

Step 3.2

Subtract $1$ from both sides of the equation.

$- x = e^{\frac{1}{4}} - 1$

Step 3.3

Divide each term in $- x = e^{\frac{1}{4}} - 1$ by $- 1$ and simplify.

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Step 3.3.1

Divide each term in $- x = e^{\frac{1}{4}} - 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{e^{\frac{1}{4}}}{- 1} + \frac{- 1}{- 1}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{e^{\frac{1}{4}}}{- 1} + \frac{- 1}{- 1}$

Step 3.3.2.2

Divide $x$ by $1$ .

$x = \frac{e^{\frac{1}{4}}}{- 1} + \frac{- 1}{- 1}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Move the negative one from the denominator of $\frac{e^{\frac{1}{4}}}{- 1}$ .

$x = - 1 \cdot e^{\frac{1}{4}} + \frac{- 1}{- 1}$

Step 3.3.3.1.2

Rewrite $- 1 \cdot e^{\frac{1}{4}}$ as $- e^{\frac{1}{4}}$ .

$x = - e^{\frac{1}{4}} + \frac{- 1}{- 1}$

Step 3.3.3.1.3

Divide $- 1$ by $- 1$ .

$x = - e^{\frac{1}{4}} + 1$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - e^{\frac{1}{4}} + 1$

Decimal Form:

$x = - 0.28402541 \dots$