Calculus Examples

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

Step 1

Simplify the right side.

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Step 1.1

Simplify $\ln (x) + \ln (\frac{2 x}{1 + x})$ .

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Step 1.1.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 1.1.2

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

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Step 1.1.2.1

Combine $x$ and $\frac{2 x}{1 + x}$ .

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

Step 1.1.2.2

Raise $x$ to the power of $1$ .

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

Step 1.1.2.3

Raise $x$ to the power of $1$ .

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

Step 1.1.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.1.2.5

Add $1$ and $1$ .

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

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

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

Step 3

Solve for $x$ .

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Step 3.1

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

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

Step 3.2

Solve the equation for $x$ .

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Step 3.2.1

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

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Step 3.2.1.1

Rewrite.

$0 + 0 + x^{2} (1 + x) = (1 - x) \cdot (2 x^{2})$

Step 3.2.1.2

Simplify by adding zeros.

$x^{2} (1 + x) = (1 - x) \cdot (2 x^{2})$

Step 3.2.1.3

Apply the distributive property.

$x^{2} \cdot 1 + x^{2} x = (1 - x) \cdot (2 x^{2})$

Step 3.2.1.4

Multiply $x^{2}$ by $1$ .

$x^{2} + x^{2} x = (1 - x) \cdot (2 x^{2})$

Step 3.2.1.5

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 3.2.1.5.1

Multiply $x^{2}$ by $x$ .

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Step 3.2.1.5.1.1

Raise $x$ to the power of $1$ .

$x^{2} + x^{2} x^{1} = (1 - x) \cdot (2 x^{2})$

Step 3.2.1.5.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2} + x^{2 + 1} = (1 - x) \cdot (2 x^{2})$

Step 3.2.1.5.2

Add $2$ and $1$ .

$x^{2} + x^{3} = (1 - x) \cdot (2 x^{2})$

Step 3.2.2

Simplify $(1 - x) \cdot (2 x^{2})$ .

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Step 3.2.2.1

Simplify by multiplying through.

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Step 3.2.2.1.1

Apply the distributive property.

$x^{2} + x^{3} = 1 (2 x^{2}) - x (2 x^{2})$

Step 3.2.2.1.2

Simplify the expression.

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Step 3.2.2.1.2.1

Multiply $2 x^{2}$ by $1$ .

$x^{2} + x^{3} = 2 x^{2} - x (2 x^{2})$

Step 3.2.2.1.2.2

Rewrite using the commutative property of multiplication.

$x^{2} + x^{3} = 2 x^{2} - 1 \cdot 2 x \cdot x^{2}$

Step 3.2.2.2

Simplify each term.

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Step 3.2.2.2.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 3.2.2.2.1.1

Move $x^{2}$ .

$x^{2} + x^{3} = 2 x^{2} - 1 \cdot 2 (x^{2} x)$

Step 3.2.2.2.1.2

Multiply $x^{2}$ by $x$ .

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Step 3.2.2.2.1.2.1

Raise $x$ to the power of $1$ .

$x^{2} + x^{3} = 2 x^{2} - 1 \cdot 2 (x^{2} x^{1})$

Step 3.2.2.2.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2} + x^{3} = 2 x^{2} - 1 \cdot 2 x^{2 + 1}$

Step 3.2.2.2.1.3

Add $2$ and $1$ .

$x^{2} + x^{3} = 2 x^{2} - 1 \cdot 2 x^{3}$

Step 3.2.2.2.2

Multiply $- 1$ by $2$ .

$x^{2} + x^{3} = 2 x^{2} - 2 x^{3}$

Step 3.2.3

Move all terms containing $x$ to the left side of the equation.

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Step 3.2.3.1

Subtract $2 x^{2}$ from both sides of the equation.

$x^{2} + x^{3} - 2 x^{2} = - 2 x^{3}$

Step 3.2.3.2

Add $2 x^{3}$ to both sides of the equation.

$x^{2} + x^{3} - 2 x^{2} + 2 x^{3} = 0$

Step 3.2.3.3

Subtract $2 x^{2}$ from $x^{2}$ .

$x^{3} - x^{2} + 2 x^{3} = 0$

Step 3.2.3.4

Add $x^{3}$ and $2 x^{3}$ .

$3 x^{3} - x^{2} = 0$

Step 3.2.4

Factor $x^{2}$ out of $3 x^{3} - x^{2}$ .

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Step 3.2.4.1

Factor $x^{2}$ out of $3 x^{3}$ .

$x^{2} (3 x) - x^{2} = 0$

Step 3.2.4.2

Factor $x^{2}$ out of $- x^{2}$ .

$x^{2} (3 x) + x^{2} \cdot - 1 = 0$

Step 3.2.4.3

Factor $x^{2}$ out of $x^{2} (3 x) + x^{2} \cdot - 1$ .

$x^{2} (3 x - 1) = 0$

Step 3.2.5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} = 0$

$3 x - 1 = 0$

Step 3.2.6

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 3.2.6.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 3.2.6.2

Solve $x^{2} = 0$ for $x$ .

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Step 3.2.6.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 3.2.6.2.2

Simplify $\pm \sqrt{0}$ .

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Step 3.2.6.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 3.2.6.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 3.2.6.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 3.2.7

Set $3 x - 1$ equal to $0$ and solve for $x$ .

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Step 3.2.7.1

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Step 3.2.7.2

Solve $3 x - 1 = 0$ for $x$ .

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Step 3.2.7.2.1

Add $1$ to both sides of the equation.

$3 x = 1$

Step 3.2.7.2.2

Divide each term in $3 x = 1$ by $3$ and simplify.

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Step 3.2.7.2.2.1

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Step 3.2.7.2.2.2

Simplify the left side.

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Step 3.2.7.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.7.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Step 3.2.7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{3}$

Step 3.2.8

The final solution is all the values that make $x^{2} (3 x - 1) = 0$ true.

$x = 0, \frac{1}{3}$

Step 4

Exclude the solutions that do not make $\ln (\frac{x^{2}}{1 - x}) = \ln (x) + \ln (\frac{2 x}{1 + x})$ true.

$x = \frac{1}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{3}$

Decimal Form:

$x = 0.\overline{3}$