Trigonometry Examples

$2 \ln (\sqrt{x}) - \ln (1 - x) = 2$

Step 1

Reorder $1$ and $- x$ .

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

Step 2

Simplify the left side.

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Step 2.1

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

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Simplify $2 \ln (\sqrt{x})$ by moving $2$ inside the logarithm.

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

Step 2.1.1.2

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 2.1.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 2.1.1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.1.1.2.3

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

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

Step 2.1.1.2.4

Cancel the common factor of $2$ .

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Step 2.1.1.2.4.1

Cancel the common factor.

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

Step 2.1.1.2.4.2

Rewrite the expression.

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

Step 2.1.1.2.5

Simplify.

$\ln (x) - \ln (- x + 1) = 2$

Step 2.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

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

Step 3

Rewrite $\ln (\frac{x}{- x + 1}) = 2$ 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 $b^{y} = x$ .

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

Step 4

Cross multiply to remove the fraction.

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

Step 5

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

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Step 5.1

Apply the distributive property.

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

Step 5.2

Simplify the expression.

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Step 5.2.1

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

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

Step 5.2.2

Reorder factors in $e^{2} (- x) + e^{2}$ .

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

Step 6

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

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

Step 7

Factor $x$ out of $x + e^{2} x$ .

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Step 7.1

Factor $x$ out of $x^{1}$ .

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

Step 7.2

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

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

Step 7.3

Factor $x$ out of $x \cdot 1 + x e^{2}$ .

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

Step 8

Divide each term in $x (1 + e^{2}) = e^{2}$ by $1 + e^{2}$ and simplify.

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Step 8.1

Divide each term in $x (1 + e^{2}) = e^{2}$ by $1 + e^{2}$ .

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

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $1 + e^{2}$ .

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $x$ by $1$ .

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.88079707 \dots$