Finite Math Examples

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

Step 1

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

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

Step 2

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

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

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.

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

Step 3.2

Solve the equation for $x$ .

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Step 3.2.1

Simplify $(x - 2) x$ .

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Step 3.2.1.1

Rewrite.

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

Step 3.2.1.2

Simplify by adding zeros.

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

Step 3.2.1.3

Apply the distributive property.

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

Step 3.2.1.4

Multiply $x$ by $x$ .

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

Step 3.2.2

Multiply $2 x + 1$ by $1$ .

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

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 $2 x$ from both sides of the equation.

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

Step 3.2.3.2

Subtract $2 x$ from $- 2 x$ .

$x^{2} - 4 x = 1$

Step 3.2.4

Subtract $1$ from both sides of the equation.

$x^{2} - 4 x - 1 = 0$

Step 3.2.5

Use the quadratic formula to find the solutions.

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

Step 3.2.6

Substitute the values $a = 1$ , $b = - 4$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

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

Step 3.2.7

Simplify.

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Step 3.2.7.1

Simplify the numerator.

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Step 3.2.7.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 3.2.7.1.2

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

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Step 3.2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 1}}{2 \cdot 1}$

Step 3.2.7.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{4 \pm \sqrt{16 + 4}}{2 \cdot 1}$

Step 3.2.7.1.3

Add $16$ and $4$ .

$x = \frac{4 \pm \sqrt{20}}{2 \cdot 1}$

Step 3.2.7.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 3.2.7.1.4.1

Factor $4$ out of $20$ .

$x = \frac{4 \pm \sqrt{4 (5)}}{2 \cdot 1}$

Step 3.2.7.1.4.2

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

$x = \frac{4 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Step 3.2.7.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{5}}{2 \cdot 1}$

Step 3.2.7.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 \sqrt{5}}{2}$

Step 3.2.7.3

Simplify $\frac{4 \pm 2 \sqrt{5}}{2}$ .

$x = 2 \pm \sqrt{5}$

Step 3.2.8

The final answer is the combination of both solutions.

$x = 2 + \sqrt{5}, 2 - \sqrt{5}$

Step 4

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

$x = 2 + \sqrt{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 2 + \sqrt{5}$

Decimal Form:

$x = 4.23606797 \dots$