Precalculus Examples

$\log_{2} (8 x) - \log_{2} (x^{2} - 1) = \log_{2} (3)$

Step 1

Simplify the left side.

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Step 1.1

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

$\log_{2} (\frac{8 x}{x^{2} - 1}) = \log_{2} (3)$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Rewrite $1$ as $1^{2}$ .

$\log_{2} (\frac{8 x}{x^{2} - 1^{2}}) = \log_{2} (3)$

Step 1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\log_{2} (\frac{8 x}{(x + 1) (x - 1)}) = \log_{2} (3)$

Step 2

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

$\frac{8 x}{(x + 1) (x - 1)} = 3$

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.

$(x + 1) (x - 1), 1$

Step 3.1.2

The LCM of one and any expression is the expression.

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

Step 3.2

Multiply each term in $\frac{8 x}{(x + 1) (x - 1)} = 3$ by $(x + 1) (x - 1)$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{8 x}{(x + 1) (x - 1)} = 3$ by $(x + 1) (x - 1)$ .

$\frac{8 x}{(x + 1) (x - 1)} ((x + 1) (x - 1)) = 3 ((x + 1) (x - 1))$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $(x + 1) (x - 1)$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{8 x}{(x + 1) (x - 1)} ((x + 1) (x - 1)) = 3 ((x + 1) (x - 1))$

Step 3.2.2.1.2

Rewrite the expression.

$8 x = 3 ((x + 1) (x - 1))$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

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

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Step 3.2.3.1.1

Apply the distributive property.

$8 x = 3 (x (x - 1) + 1 (x - 1))$

Step 3.2.3.1.2

Apply the distributive property.

$8 x = 3 (x \cdot x + x \cdot - 1 + 1 (x - 1))$

Step 3.2.3.1.3

Apply the distributive property.

$8 x = 3 (x \cdot x + x \cdot - 1 + 1 x + 1 \cdot - 1)$

Step 3.2.3.2

Simplify and combine like terms.

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Step 3.2.3.2.1

Simplify each term.

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Step 3.2.3.2.1.1

Multiply $x$ by $x$ .

$8 x = 3 (x^{2} + x \cdot - 1 + 1 x + 1 \cdot - 1)$

Step 3.2.3.2.1.2

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

$8 x = 3 (x^{2} - 1 \cdot x + 1 x + 1 \cdot - 1)$

Step 3.2.3.2.1.3

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

$8 x = 3 (x^{2} - x + 1 x + 1 \cdot - 1)$

Step 3.2.3.2.1.4

Multiply $x$ by $1$ .

$8 x = 3 (x^{2} - x + x + 1 \cdot - 1)$

Step 3.2.3.2.1.5

Multiply $- 1$ by $1$ .

$8 x = 3 (x^{2} - x + x - 1)$

Step 3.2.3.2.2

Add $- x$ and $x$ .

$8 x = 3 (x^{2} + 0 - 1)$

Step 3.2.3.2.3

Add $x^{2}$ and $0$ .

$8 x = 3 (x^{2} - 1)$

Step 3.2.3.3

Apply the distributive property.

$8 x = 3 x^{2} + 3 \cdot - 1$

Step 3.2.3.4

Multiply $3$ by $- 1$ .

$8 x = 3 x^{2} - 3$

Step 3.3

Solve the equation.

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Step 3.3.1

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

$8 x - 3 x^{2} = - 3$

Step 3.3.2

Add $3$ to both sides of the equation.

$8 x - 3 x^{2} + 3 = 0$

Step 3.3.3

Factor the left side of the equation.

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Step 3.3.3.1

Factor $- 1$ out of $8 x - 3 x^{2} + 3$ .

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Step 3.3.3.1.1

Reorder $8 x$ and $- 3 x^{2}$ .

$- 3 x^{2} + 8 x + 3 = 0$

Step 3.3.3.1.2

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

$- (3 x^{2}) + 8 x + 3 = 0$

Step 3.3.3.1.3

Factor $- 1$ out of $8 x$ .

$- (3 x^{2}) - (- 8 x) + 3 = 0$

Step 3.3.3.1.4

Rewrite $3$ as $- 1 (- 3)$ .

$- (3 x^{2}) - (- 8 x) - 1 \cdot - 3 = 0$

Step 3.3.3.1.5

Factor $- 1$ out of $- (3 x^{2}) - (- 8 x)$ .

$- (3 x^{2} - 8 x) - 1 \cdot - 3 = 0$

Step 3.3.3.1.6

Factor $- 1$ out of $- (3 x^{2} - 8 x) - 1 (- 3)$ .

$- (3 x^{2} - 8 x - 3) = 0$

Step 3.3.3.2

Factor.

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Step 3.3.3.2.1

Factor by grouping.

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Step 3.3.3.2.1.1

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 $a \cdot c = 3 \cdot - 3 = - 9$ and whose sum is $b = - 8$ .

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Step 3.3.3.2.1.1.1

Factor $- 8$ out of $- 8 x$ .

$- (3 x^{2} - 8 x - 3) = 0$

Step 3.3.3.2.1.1.2

Rewrite $- 8$ as $1$ plus $- 9$

$- (3 x^{2} + (1 - 9) x - 3) = 0$

Step 3.3.3.2.1.1.3

Apply the distributive property.

$- (3 x^{2} + 1 x - 9 x - 3) = 0$

Step 3.3.3.2.1.1.4

Multiply $x$ by $1$ .

$- (3 x^{2} + x - 9 x - 3) = 0$

Step 3.3.3.2.1.2

Factor out the greatest common factor from each group.

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Step 3.3.3.2.1.2.1

Group the first two terms and the last two terms.

$- ((3 x^{2} + x) - 9 x - 3) = 0$

Step 3.3.3.2.1.2.2

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

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

Step 3.3.3.2.1.3

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

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

Step 3.3.3.2.2

Remove unnecessary parentheses.

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

Step 3.3.4

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

$3 x + 1 = 0$

$x - 3 = 0$

Step 3.3.5

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

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Step 3.3.5.1

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

$3 x + 1 = 0$

Step 3.3.5.2

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

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Step 3.3.5.2.1

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Step 3.3.5.2.2

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

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Step 3.3.5.2.2.1

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

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

Step 3.3.5.2.2.2

Simplify the left side.

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Step 3.3.5.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.3.5.2.2.2.1.1

Cancel the common factor.

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

Step 3.3.5.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.3.5.2.2.3

Simplify the right side.

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Step 3.3.5.2.2.3.1

Move the negative in front of the fraction.

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

Step 3.3.6

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

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Step 3.3.6.1

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

$x - 3 = 0$

Step 3.3.6.2

Add $3$ to both sides of the equation.

$x = 3$

Step 3.3.7

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

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

Step 4

Exclude the solutions that do not make $\log_{2} (8 x) - \log_{2} (x^{2} - 1) = \log_{2} (3)$ true.

$x = 3$