Algebra Examples

$\log_{9} (x - 4) + \log_{9} (x + 1) = \log_{9} (x + 5) + \log_{9} (x - 6)$

Step 1

Simplify the left side.

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Step 1.1

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

$\log_{9} ((x - 4) (x + 1)) = \log_{9} (x + 5) + \log_{9} (x - 6)$

Step 1.2

Expand $(x - 4) (x + 1)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\log_{9} (x (x + 1) - 4 (x + 1)) = \log_{9} (x + 5) + \log_{9} (x - 6)$

Step 1.2.2

Apply the distributive property.

$\log_{9} (x \cdot x + x \cdot 1 - 4 (x + 1)) = \log_{9} (x + 5) + \log_{9} (x - 6)$

Step 1.2.3

Apply the distributive property.

$\log_{9} (x \cdot x + x \cdot 1 - 4 x - 4 \cdot 1) = \log_{9} (x + 5) + \log_{9} (x - 6)$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

$\log_{9} (x^{2} + x \cdot 1 - 4 x - 4 \cdot 1) = \log_{9} (x + 5) + \log_{9} (x - 6)$

Step 1.3.1.2

Multiply $x$ by $1$ .

$\log_{9} (x^{2} + x - 4 x - 4 \cdot 1) = \log_{9} (x + 5) + \log_{9} (x - 6)$

Step 1.3.1.3

Multiply $- 4$ by $1$ .

$\log_{9} (x^{2} + x - 4 x - 4) = \log_{9} (x + 5) + \log_{9} (x - 6)$

Step 1.3.2

Subtract $4 x$ from $x$ .

$\log_{9} (x^{2} - 3 x - 4) = \log_{9} (x + 5) + \log_{9} (x - 6)$

Step 2

Simplify the right side.

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Step 2.1

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

$\log_{9} (x^{2} - 3 x - 4) = \log_{9} ((x + 5) (x - 6))$

Step 2.2

Expand $(x + 5) (x - 6)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$\log_{9} (x^{2} - 3 x - 4) = \log_{9} (x (x - 6) + 5 (x - 6))$

Step 2.2.2

Apply the distributive property.

$\log_{9} (x^{2} - 3 x - 4) = \log_{9} (x \cdot x + x \cdot - 6 + 5 (x - 6))$

Step 2.2.3

Apply the distributive property.

$\log_{9} (x^{2} - 3 x - 4) = \log_{9} (x \cdot x + x \cdot - 6 + 5 x + 5 \cdot - 6)$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $x$ by $x$ .

$\log_{9} (x^{2} - 3 x - 4) = \log_{9} (x^{2} + x \cdot - 6 + 5 x + 5 \cdot - 6)$

Step 2.3.1.2

Move $- 6$ to the left of $x$ .

$\log_{9} (x^{2} - 3 x - 4) = \log_{9} (x^{2} - 6 \cdot x + 5 x + 5 \cdot - 6)$

Step 2.3.1.3

Multiply $5$ by $- 6$ .

$\log_{9} (x^{2} - 3 x - 4) = \log_{9} (x^{2} - 6 x + 5 x - 30)$

Step 2.3.2

Add $- 6 x$ and $5 x$ .

$\log_{9} (x^{2} - 3 x - 4) = \log_{9} (x^{2} - x - 30)$

Step 3

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

$x^{2} - 3 x - 4 = x^{2} - x - 30$

Step 4

Solve for $x$ .

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Step 4.1

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

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Step 4.1.1

Subtract $x^{2}$ from both sides of the equation.

$x^{2} - 3 x - 4 - x^{2} = - x - 30$

Step 4.1.2

Add $x$ to both sides of the equation.

$x^{2} - 3 x - 4 - x^{2} + x = - 30$

Step 4.1.3

Combine the opposite terms in $x^{2} - 3 x - 4 - x^{2} + x$ .

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Step 4.1.3.1

Subtract $x^{2}$ from $x^{2}$ .

$- 3 x - 4 + 0 + x = - 30$

Step 4.1.3.2

Add $- 3 x - 4$ and $0$ .

$- 3 x - 4 + x = - 30$

Step 4.1.4

Add $- 3 x$ and $x$ .

$- 2 x - 4 = - 30$

Step 4.2

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

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Step 4.2.1

Add $4$ to both sides of the equation.

$- 2 x = - 30 + 4$

Step 4.2.2

Add $- 30$ and $4$ .

$- 2 x = - 26$

Step 4.3

Divide each term in $- 2 x = - 26$ by $- 2$ and simplify.

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Step 4.3.1

Divide each term in $- 2 x = - 26$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 26}{- 2}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $- 2$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 26}{- 2}$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 26}{- 2}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide $- 26$ by $- 2$ .

$x = 13$