Calculus Examples

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

Step 1

Multiply the equation by $x - 1$ .

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

Step 2

Simplify the left side.

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Step 2.1

Simplify $(x - 1) (y)$ .

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Step 2.1.1

Apply the distributive property.

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

Step 2.1.2

Rewrite $- 1 y$ as $- y$ .

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

Step 3

Simplify the right side.

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Step 3.1

Simplify $(\ln (\frac{x}{x - 1})) (x - 1)$ .

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Step 3.1.1

Simplify by multiplying through.

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Step 3.1.1.1

Apply the distributive property.

$x y - y = \ln (\frac{x}{x - 1}) x + \ln (\frac{x}{x - 1}) \cdot - 1$

Step 3.1.1.2

Move $- 1$ to the left of $\ln (\frac{x}{x - 1})$ .

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

Step 3.1.2

Rewrite $- 1 \ln (\frac{x}{x - 1})$ as $- \ln (\frac{x}{x - 1})$ .

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

Step 3.1.3

Reorder factors in $\ln (\frac{x}{x - 1}) x - \ln (\frac{x}{x - 1})$ .

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

Step 4

Move all the terms containing a logarithm to the left side of the equation.

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

Step 5

Rewrite the equation as $\ln (\frac{x}{x - 1}) = y$ .

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

Step 6

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

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

Step 7

Rewrite $\ln (\frac{x}{x - 1}) = y$ 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 $b^{y} = x$ .

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

Step 8

Solve for $x$ .

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Step 8.1

Multiply the equation by $x - 1$ .

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

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Apply the distributive property.

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

Step 8.2.1.2

Rewrite $- 1 e^{y}$ as $- e^{y}$ .

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

Step 8.3

Simplify the right side.

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Step 8.3.1

Cancel the common factor of $x - 1$ .

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Step 8.3.1.1

Cancel the common factor.

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

Step 8.3.1.2

Rewrite the expression.

$x e^{y} - e^{y} = x$

Step 8.4

Solve for $x$ .

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Step 8.4.1

Subtract $x$ from both sides of the equation.

$x e^{y} - e^{y} - x = 0$

Step 8.4.2

Add $e^{y}$ to both sides of the equation.

$x e^{y} - x = e^{y}$

Step 8.4.3

Factor $x$ out of $x e^{y} - x$ .

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Step 8.4.3.1

Factor $x$ out of $x e^{y}$ .

$x (e^{y}) - x = e^{y}$

Step 8.4.3.2

Factor $x$ out of $- x$ .

$x (e^{y}) + x \cdot - 1 = e^{y}$

Step 8.4.3.3

Factor $x$ out of $x (e^{y}) + x \cdot - 1$ .

$x (e^{y} - 1) = e^{y}$

Step 8.4.4

Divide each term in $x (e^{y} - 1) = e^{y}$ by $e^{y} - 1$ and simplify.

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Step 8.4.4.1

Divide each term in $x (e^{y} - 1) = e^{y}$ by $e^{y} - 1$ .

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

Step 8.4.4.2

Simplify the left side.

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Step 8.4.4.2.1

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

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Step 8.4.4.2.1.1

Cancel the common factor.

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

Step 8.4.4.2.1.2

Divide $x$ by $1$ .

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