Precalculus Examples

$\log (x^{2} - 1) - \log (12) = 1$

Step 1

Simplify the left side.

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Step 1.1

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

$\log (\frac{x^{2} - 1}{12}) = 1$

Step 1.2

Simplify the numerator.

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Step 1.2.1

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

$\log (\frac{x^{2} - 1^{2}}{12}) = 1$

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 $b = 1$ .

$\log (\frac{(x + 1) (x - 1)}{12}) = 1$

Step 2

Rewrite $\log (\frac{(x + 1) (x - 1)}{12}) = 1$ 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 $b^{y} = x$ .

$10^{1} = \frac{(x + 1) (x - 1)}{12}$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $\frac{(x + 1) (x - 1)}{12} = 10^{1}$ .

$\frac{(x + 1) (x - 1)}{12} = 10$

Step 3.2

Multiply both sides of the equation by $12$ .

$12 \frac{(x + 1) (x - 1)}{12} = 12 \cdot 10^{1}$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $12 \frac{(x + 1) (x - 1)}{12}$ .

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Step 3.3.1.1.1

Cancel the common factor of $12$ .

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Step 3.3.1.1.1.1

Cancel the common factor.

$12 \frac{(x + 1) (x - 1)}{12} = 12 \cdot 10^{1}$

Step 3.3.1.1.1.2

Rewrite the expression.

$(x + 1) (x - 1) = 12 \cdot 10^{1}$

Step 3.3.1.1.2

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

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Step 3.3.1.1.2.1

Apply the distributive property.

$x (x - 1) + 1 (x - 1) = 12 \cdot 10^{1}$

Step 3.3.1.1.2.2

Apply the distributive property.

$x \cdot x + x \cdot - 1 + 1 (x - 1) = 12 \cdot 10^{1}$

Step 3.3.1.1.2.3

Apply the distributive property.

$x \cdot x + x \cdot - 1 + 1 x + 1 \cdot - 1 = 12 \cdot 10^{1}$

Step 3.3.1.1.3

Simplify and combine like terms.

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Step 3.3.1.1.3.1

Simplify each term.

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Step 3.3.1.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 1 + 1 x + 1 \cdot - 1 = 12 \cdot 10^{1}$

Step 3.3.1.1.3.1.2

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

$x^{2} - 1 \cdot x + 1 x + 1 \cdot - 1 = 12 \cdot 10^{1}$

Step 3.3.1.1.3.1.3

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

$x^{2} - x + 1 x + 1 \cdot - 1 = 12 \cdot 10^{1}$

Step 3.3.1.1.3.1.4

Multiply $x$ by $1$ .

$x^{2} - x + x + 1 \cdot - 1 = 12 \cdot 10^{1}$

Step 3.3.1.1.3.1.5

Multiply $- 1$ by $1$ .

$x^{2} - x + x - 1 = 12 \cdot 10^{1}$

Step 3.3.1.1.3.2

Add $- x$ and $x$ .

$x^{2} + 0 - 1 = 12 \cdot 10^{1}$

Step 3.3.1.1.3.3

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

$x^{2} - 1 = 12 \cdot 10^{1}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Move the decimal point in $12$ to the left by $1$ place and increase the power of $10^{1}$ by $1$ .

$x^{2} - 1 = 1.2 \cdot 10^{2}$

Step 3.4

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

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Step 3.4.1

Add $1$ to both sides of the equation.

$x^{2} = 1.2 \cdot 10^{2} + 1$

Step 3.4.2

Convert $1$ to scientific notation.

$x^{2} = 1.2 \cdot 10^{2} + 1 \cdot 10^{0}$

Step 3.4.3

Move the decimal point in $1$ to the left by $2$ places and increase the power of $10^{0}$ by $2$ .

$x^{2} = 1.2 \cdot 10^{2} + 0.01 \cdot 10^{2}$

Step 3.4.4

Factor $10^{2}$ out of $1.2 \cdot 10^{2} + 0.01 \cdot 10^{2}$ .

$x^{2} = (1.2 + 0.01) \cdot 10^{2}$

Step 3.4.5

Add $1.2$ and $0.01$ .

$x^{2} = 1.21 \cdot 10^{2}$

Step 3.5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{1.21 \cdot 10^{2}}$

Step 3.6

Simplify $\pm \sqrt{1.21 \cdot 10^{2}}$ .

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Step 3.6.1

Rewrite $\sqrt{1.21 \cdot 10^{2}}$ as $\sqrt{1.21} \cdot \sqrt{10^{2}}$ .

$x = \pm \sqrt{1.21} \cdot \sqrt{10^{2}}$

Step 3.6.2

Evaluate the root.

$x = \pm 1.1 \cdot \sqrt{10^{2}}$

Step 3.6.3

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 1.1 \cdot 10$

Step 3.6.4

Raise $10$ to the power of $1$ .

$x = \pm 1.1 \cdot 10^{1}$

Step 3.7

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.7.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 1.1 \cdot 10^{1}$

Step 3.7.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 1.1 \cdot 10^{1}$

Step 3.7.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 1.1 \cdot 10^{1}, - 1.1 \cdot 10^{1}$

Step 4

The result can be shown in multiple forms.

Scientific Notation:

$x = 1.1 \cdot 10^{1}, - 1.1 \cdot 10^{1}$

Expanded Form:

$x = 11, - 11$