Trigonometry Examples

$\log_{5} (x + 1) - \log_{5} (x - 1) = \log_{5} (x)$

Step 1

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

$\log_{5} (\frac{x + 1}{x - 1}) = \log_{5} (x)$

Step 2

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

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

Step 3

Solve for $x$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x - 1, 1$

Step 3.1.2

Remove parentheses.

$x - 1, 1$

Step 3.1.3

The LCM of one and any expression is the expression.

$x - 1$

Step 3.2

Multiply each term in $\frac{x + 1}{x - 1} = x$ by $x - 1$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{x + 1}{x - 1} = x$ by $x - 1$ .

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $x - 1$ .

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Step 3.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.1.2

Rewrite the expression.

$x + 1 = x (x - 1)$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify by multiplying through.

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Step 3.2.3.1.1

Apply the distributive property.

$x + 1 = x \cdot x + x \cdot - 1$

Step 3.2.3.1.2

Simplify the expression.

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Step 3.2.3.1.2.1

Multiply $x$ by $x$ .

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

Step 3.2.3.1.2.2

Move $- 1$ to the left of $x$ .

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

Step 3.2.3.2

Rewrite $- 1 x$ as $- x$ .

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

Step 3.3

Solve the equation.

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Step 3.3.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 3.3.2

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

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Step 3.3.2.1

Subtract $x$ from both sides of the equation.

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

Step 3.3.2.2

Subtract $x$ from $- x$ .

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

Step 3.3.3

Subtract $1$ from both sides of the equation.

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

Step 3.3.4

Use the quadratic formula to find the solutions.

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

Step 3.3.5

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

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

Step 3.3.6

Simplify.

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Step 3.3.6.1

Simplify the numerator.

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Step 3.3.6.1.1

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

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

Step 3.3.6.1.2

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

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Step 3.3.6.1.2.1

Multiply $- 4$ by $1$ .

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

Step 3.3.6.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 3.3.6.1.3

Add $4$ and $4$ .

$x = \frac{2 \pm \sqrt{8}}{2 \cdot 1}$

Step 3.3.6.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.3.6.1.4.1

Factor $4$ out of $8$ .

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

Step 3.3.6.1.4.2

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

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

Step 3.3.6.1.5

Pull terms out from under the radical.

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

Step 3.3.6.2

Multiply $2$ by $1$ .

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

Step 3.3.6.3

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

$x = 1 \pm \sqrt{2}$

Step 3.3.7

The final answer is the combination of both solutions.

$x = 1 + \sqrt{2}, 1 - \sqrt{2}$

Step 4

Exclude the solutions that do not make $\log_{5} (x + 1) - \log_{5} (x - 1) = \log_{5} (x)$ true.

$x = 1 + \sqrt{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 1 + \sqrt{2}$

Decimal Form:

$x = 2.41421356 \dots$