Precalculus Examples

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

Step 1

Move all the terms containing a logarithm to the left side of the equation.

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

Step 2

Simplify the left side.

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Step 2.1

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

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Step 2.1.1

Simplify $- 2 \log_{3} (x - 1)$ by moving $2$ inside the logarithm.

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

Step 2.1.2

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

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

Step 3

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

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

Step 4

Cross multiply to remove the fraction.

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

Step 5

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

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Step 5.1

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

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

Step 5.2

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

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Step 5.2.1

Apply the distributive property.

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

Step 5.2.2

Apply the distributive property.

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

Step 5.2.3

Apply the distributive property.

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

Step 5.3

Simplify and combine like terms.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply $x$ by $x$ .

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

Step 5.3.1.2

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

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

Step 5.3.1.3

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

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

Step 5.3.1.4

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

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

Step 5.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 5.3.2

Subtract $x$ from $- x$ .

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

Step 5.4

Apply the distributive property.

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

Step 5.5

Simplify.

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Step 5.5.1

Multiply $- 2$ by $3$ .

$x + 1 = 3 x^{2} - 6 x + 3 \cdot 1$

Step 5.5.2

Multiply $3$ by $1$ .

$x + 1 = 3 x^{2} - 6 x + 3$

Step 6

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

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Step 6.1

Subtract $3 x^{2}$ from both sides of the equation.

$x + 1 - 3 x^{2} = - 6 x + 3$

Step 6.2

Add $6 x$ to both sides of the equation.

$x + 1 - 3 x^{2} + 6 x = 3$

Step 6.3

Add $x$ and $6 x$ .

$7 x + 1 - 3 x^{2} = 3$

Step 7

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

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Step 7.1

Subtract $1$ from both sides of the equation.

$7 x - 3 x^{2} = 3 - 1$

Step 7.2

Subtract $1$ from $3$ .

$7 x - 3 x^{2} = 2$

Step 8

Subtract $2$ from both sides of the equation.

$7 x - 3 x^{2} - 2 = 0$

Step 9

Factor the left side of the equation.

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Step 9.1

Factor $- 1$ out of $7 x - 3 x^{2} - 2$ .

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Step 9.1.1

Reorder $7 x$ and $- 3 x^{2}$ .

$- 3 x^{2} + 7 x - 2 = 0$

Step 9.1.2

Factor $- 1$ out of $- 3 x^{2}$ .

$- (3 x^{2}) + 7 x - 2 = 0$

Step 9.1.3

Factor $- 1$ out of $7 x$ .

$- (3 x^{2}) - (- 7 x) - 2 = 0$

Step 9.1.4

Rewrite $- 2$ as $- 1 (2)$ .

$- (3 x^{2}) - (- 7 x) - 1 \cdot 2 = 0$

Step 9.1.5

Factor $- 1$ out of $- (3 x^{2}) - (- 7 x)$ .

$- (3 x^{2} - 7 x) - 1 \cdot 2 = 0$

Step 9.1.6

Factor $- 1$ out of $- (3 x^{2} - 7 x) - 1 (2)$ .

$- (3 x^{2} - 7 x + 2) = 0$

Step 9.2

Factor.

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Step 9.2.1

Factor by grouping.

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Step 9.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 2 = 6$ and whose sum is $b = - 7$ .

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Step 9.2.1.1.1

Factor $- 7$ out of $- 7 x$ .

$- (3 x^{2} - 7 x + 2) = 0$

Step 9.2.1.1.2

Rewrite $- 7$ as $- 1$ plus $- 6$

$- (3 x^{2} + (- 1 - 6) x + 2) = 0$

Step 9.2.1.1.3

Apply the distributive property.

$- (3 x^{2} - 1 x - 6 x + 2) = 0$

Step 9.2.1.2

Factor out the greatest common factor from each group.

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Step 9.2.1.2.1

Group the first two terms and the last two terms.

$- ((3 x^{2} - 1 x) - 6 x + 2) = 0$

Step 9.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$- (x (3 x - 1) - 2 (3 x - 1)) = 0$

Step 9.2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$- ((3 x - 1) (x - 2)) = 0$

Step 9.2.2

Remove unnecessary parentheses.

$- (3 x - 1) (x - 2) = 0$

Step 10

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$3 x - 1 = 0$

$x - 2 = 0$

Step 11

Set $3 x - 1$ equal to $0$ and solve for $x$ .

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Step 11.1

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Step 11.2

Solve $3 x - 1 = 0$ for $x$ .

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Step 11.2.1

Add $1$ to both sides of the equation.

$3 x = 1$

Step 11.2.2

Divide each term in $3 x = 1$ by $3$ and simplify.

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Step 11.2.2.1

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Step 11.2.2.2

Simplify the left side.

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Step 11.2.2.2.1

Cancel the common factor of $3$ .

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Step 11.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Step 11.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{3}$

Step 12

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 12.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 12.2

Add $2$ to both sides of the equation.

$x = 2$

Step 13

The final solution is all the values that make $- (3 x - 1) (x - 2) = 0$ true.

$x = \frac{1}{3}, 2$

Step 14

Exclude the solutions that do not make $\log_{3} (x + 1) - 2 \log_{3} (x - 1) = 1$ true.

$x = 2$