Trigonometry Examples

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

Step 1

Simplify the left side.

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

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

Apply the distributive property.

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

Simplify the expression.

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Multiply $x$ by $x$ .

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

Move $- 2$ to the left of $x$ .

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

Step 2

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

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

Step 3

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

$e^{1} = x^{2} - 2 x$

Step 4

Solve for $x$ .

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Rewrite the equation as $x^{2} - 2 x = e^{1}$ .

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

Subtract $e$ from both sides of the equation.

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

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = - 2$ , and $c = - e$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot (- e))}}{2 \cdot 1}$

Simplify.

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Simplify the numerator.

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Raise $- 2$ to the power of $2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot (- 1 e)}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot (- 1 e)}}{2 \cdot 1}$

Multiply $- 4$ by $- 1$ .

$x = \frac{2 \pm \sqrt{4 + 4 e}}{2 \cdot 1}$

Rewrite $4 + 4 e$ as $2^{2} (1 + e)$ .

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Factor $4$ out of $4$ .

$x = \frac{2 \pm \sqrt{4 (1) + 4 e}}{2 \cdot 1}$

Factor $4$ out of $4 e$ .

$x = \frac{2 \pm \sqrt{4 (1) + 4 (e)}}{2 \cdot 1}$

Factor $4$ out of $4 (1) + 4 (e)$ .

$x = \frac{2 \pm \sqrt{4 (1 + e)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

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

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{1 + e}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 \sqrt{1 + e}}{2}$

Simplify $\frac{2 \pm 2 \sqrt{1 + e}}{2}$ .

$x = 1 \pm \sqrt{1 + e}$

The final answer is the combination of both solutions.

$x = 1 + \sqrt{1 + e}, 1 - \sqrt{1 + e}$

Step 5

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

$x = 1 + \sqrt{1 + e}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = 1 + \sqrt{1 + e}$

Decimal Form:

$x = 2.92828468 \dots$