Trigonometry Examples

$\log_{9} (x + 5) - \log_{9} (x - 5) = - 1$

Step 1

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

$\log_{9} (\frac{x + 5}{x - 5}) = - 1$

Step 2

Rewrite $\log_{9} (\frac{x + 5}{x - 5}) = - 1$ 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 $b^{y} = x$ .

$9^{- 1} = \frac{x + 5}{x - 5}$

Step 3

Cross multiply to remove the fraction.

$x + 5 = 9^{- 1} (x - 5)$

Step 4

Simplify $9^{- 1} (x - 5)$ .

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Step 4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x + 5 = \frac{1}{9} (x - 5)$

Step 4.2

Apply the distributive property.

$x + 5 = \frac{1}{9} x + \frac{1}{9} \cdot - 5$

Step 4.3

Combine $\frac{1}{9}$ and $x$ .

$x + 5 = \frac{x}{9} + \frac{1}{9} \cdot - 5$

Step 4.4

Combine $\frac{1}{9}$ and $- 5$ .

$x + 5 = \frac{x}{9} + \frac{- 5}{9}$

Step 4.5

Move the negative in front of the fraction.

$x + 5 = \frac{x}{9} - \frac{5}{9}$

Step 5

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

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Step 5.1

Subtract $\frac{x}{9}$ from both sides of the equation.

$x + 5 - \frac{x}{9} = - \frac{5}{9}$

Step 5.2

To write $x$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$x \cdot \frac{9}{9} - \frac{x}{9} + 5 = - \frac{5}{9}$

Step 5.3

Combine $x$ and $\frac{9}{9}$ .

$\frac{x \cdot 9}{9} - \frac{x}{9} + 5 = - \frac{5}{9}$

Step 5.4

Combine the numerators over the common denominator.

$\frac{x \cdot 9 - x}{9} + 5 = - \frac{5}{9}$

Step 5.5

Subtract $x$ from $x \cdot 9$ .

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Step 5.5.1

Reorder $x$ and $9$ .

$\frac{9 \cdot x - x}{9} + 5 = - \frac{5}{9}$

Step 5.5.2

Subtract $x$ from $9 \cdot x$ .

$\frac{8 \cdot x}{9} + 5 = - \frac{5}{9}$

$\frac{8 x}{9} + 5 = - \frac{5}{9}$

Step 6

Move all terms not containing $x$ to the right side of the equation.

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Step 6.1

Subtract $5$ from both sides of the equation.

$\frac{8 x}{9} = - \frac{5}{9} - 5$

Step 6.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{8 x}{9} = - \frac{5}{9} - 5 \cdot \frac{9}{9}$

Step 6.3

Combine $- 5$ and $\frac{9}{9}$ .

$\frac{8 x}{9} = - \frac{5}{9} + \frac{- 5 \cdot 9}{9}$

Step 6.4

Combine the numerators over the common denominator.

$\frac{8 x}{9} = \frac{- 5 - 5 \cdot 9}{9}$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Multiply $- 5$ by $9$ .

$\frac{8 x}{9} = \frac{- 5 - 45}{9}$

Step 6.5.2

Subtract $45$ from $- 5$ .

$\frac{8 x}{9} = \frac{- 50}{9}$

Step 6.6

Move the negative in front of the fraction.

$\frac{8 x}{9} = - \frac{50}{9}$

Step 7

Multiply both sides by $9$ .

$\frac{8 x}{9} \cdot 9 = - \frac{50}{9} \cdot 9$

Step 8

Simplify.

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Step 8.1

Simplify the left side.

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Step 8.1.1

Cancel the common factor of $9$ .

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Step 8.1.1.1

Cancel the common factor.

$\frac{8 x}{9} \cdot 9 = - \frac{50}{9} \cdot 9$

Step 8.1.1.2

Rewrite the expression.

$8 x = - \frac{50}{9} \cdot 9$

Step 8.2

Simplify the right side.

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Step 8.2.1

Cancel the common factor of $9$ .

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Step 8.2.1.1

Move the leading negative in $- \frac{50}{9}$ into the numerator.

$8 x = \frac{- 50}{9} \cdot 9$

Step 8.2.1.2

Cancel the common factor.

$8 x = \frac{- 50}{9} \cdot 9$

Step 8.2.1.3

Rewrite the expression.

$8 x = - 50$

Step 9

Divide each term in $8 x = - 50$ by $8$ and simplify.

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Step 9.1

Divide each term in $8 x = - 50$ by $8$ .

$\frac{8 x}{8} = \frac{- 50}{8}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $8$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{- 50}{8}$

Step 9.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 50}{8}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Cancel the common factor of $- 50$ and $8$ .

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Step 9.3.1.1

Factor $2$ out of $- 50$ .

$x = \frac{2 (- 25)}{8}$

Step 9.3.1.2

Cancel the common factors.

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Step 9.3.1.2.1

Factor $2$ out of $8$ .

$x = \frac{2 \cdot - 25}{2 \cdot 4}$

Step 9.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot - 25}{2 \cdot 4}$

Step 9.3.1.2.3

Rewrite the expression.

$x = \frac{- 25}{4}$

Step 9.3.2

Move the negative in front of the fraction.

$x = - \frac{25}{4}$

Step 10

Exclude the solutions that do not make $\log_{9} (x + 5) - \log_{9} (x - 5) = - 1$ true.

$No solution$