Algebra Examples

$\log_{3} (3 x + 3) - \log_{3} (x - 1) = 2$

Step 1

Simplify the left side.

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Step 1.1

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

$\log_{3} (\frac{3 x + 3}{x - 1}) = 2$

Step 1.2

Factor $3$ out of $3 x + 3$ .

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Step 1.2.1

Factor $3$ out of $3 x$ .

$\log_{3} (\frac{3 (x) + 3}{x - 1}) = 2$

Step 1.2.2

Factor $3$ out of $3$ .

$\log_{3} (\frac{3 (x) + 3 (1)}{x - 1}) = 2$

Step 1.2.3

Factor $3$ out of $3 (x) + 3 (1)$ .

$\log_{3} (\frac{3 (x + 1)}{x - 1}) = 2$

Step 2

Rewrite $\log_{3} (\frac{3 (x + 1)}{x - 1}) = 2$ 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 $b^{y} = x$ .

$3^{2} = \frac{3 (x + 1)}{x - 1}$

Step 3

Cross multiply to remove the fraction.

$3 (x + 1) = 3^{2} (x - 1)$

Step 4

Simplify $3^{2} (x - 1)$ .

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Step 4.1

Raise $3$ to the power of $2$ .

$3 (x + 1) = 9 (x - 1)$

Step 4.2

Apply the distributive property.

$3 (x + 1) = 9 x + 9 \cdot - 1$

Step 4.3

Multiply $9$ by $- 1$ .

$3 (x + 1) = 9 x - 9$

Step 5

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

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Step 5.1

Subtract $9 x$ from both sides of the equation.

$3 (x + 1) - 9 x = - 9$

Step 5.2

Simplify each term.

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Step 5.2.1

Apply the distributive property.

$3 x + 3 \cdot 1 - 9 x = - 9$

Step 5.2.2

Multiply $3$ by $1$ .

$3 x + 3 - 9 x = - 9$

Step 5.3

Subtract $9 x$ from $3 x$ .

$- 6 x + 3 = - 9$

Step 6

Factor $3$ out of $- 6 x + 3$ .

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Step 6.1

Factor $3$ out of $- 6 x$ .

$3 (- 2 x) + 3 = - 9$

Step 6.2

Factor $3$ out of $3$ .

$3 (- 2 x) + 3 (1) = - 9$

Step 6.3

Factor $3$ out of $3 (- 2 x) + 3 (1)$ .

$3 (- 2 x + 1) = - 9$

Step 7

Divide each term in $3 (- 2 x + 1) = - 9$ by $3$ and simplify.

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Step 7.1

Divide each term in $3 (- 2 x + 1) = - 9$ by $3$ .

$\frac{3 (- 2 x + 1)}{3} = \frac{- 9}{3}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $3$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{3 (- 2 x + 1)}{3} = \frac{- 9}{3}$

Step 7.2.1.2

Divide $- 2 x + 1$ by $1$ .

$- 2 x + 1 = \frac{- 9}{3}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $- 9$ by $3$ .

$- 2 x + 1 = - 3$

Step 8

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

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Step 8.1

Subtract $1$ from both sides of the equation.

$- 2 x = - 3 - 1$

Step 8.2

Subtract $1$ from $- 3$ .

$- 2 x = - 4$

Step 9

Divide each term in $- 2 x = - 4$ by $- 2$ and simplify.

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Step 9.1

Divide each term in $- 2 x = - 4$ by $- 2$ .

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

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $- 2$ .

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Step 9.2.1.1

Cancel the common factor.

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

Step 9.2.1.2

Divide $x$ by $1$ .

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

Step 9.3

Simplify the right side.

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Step 9.3.1

Divide $- 4$ by $- 2$ .

$x = 2$