Trigonometry Examples

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

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

Step 1

Reorder

1

$1$ and

- x

$- x$ .

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

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

Step 2

Simplify the left side.

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Step 2.1

Simplify

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

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

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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})

$2 \ln (\sqrt{x})$ by moving

2

$2$ inside the logarithm.

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

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

Step 2.1.1.2

Rewrite

{\sqrt{x}}^{2}

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

x

$x$ .

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Step 2.1.1.2.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{x}

$\sqrt{x}$ as

x^{\frac{1}{2}}

$x^{\frac{1}{2}}$ .

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

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 2.1.1.2.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 2.1.1.2.4

Cancel the common factor of

2

$2$ .

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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:

$x = 0.88079707 \dots$