Algebra Examples

$\log (x^{2} - 2) + 2 \log (6) = \log (6 x)$

Step 1

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

$\log (x^{2} - 2) + 2 \log (6) - \log (6 x) = 0$

Step 2

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

$2 \log (6) + \log (\frac{x^{2} - 2}{6 x}) = 0$

Step 3

Simplify the left side.

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Step 3.1

Simplify $2 \log (6) + \log (\frac{x^{2} - 2}{6 x})$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Simplify $2 \log (6)$ by moving $2$ inside the logarithm.

$\log (6^{2}) + \log (\frac{x^{2} - 2}{6 x}) = 0$

Step 3.1.1.2

Raise $6$ to the power of $2$ .

$\log (36) + \log (\frac{x^{2} - 2}{6 x}) = 0$

Step 3.1.2

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

$\log (36 \frac{x^{2} - 2}{6 x}) = 0$

Step 3.1.3

Cancel the common factor of $6$ .

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Step 3.1.3.1

Factor $6$ out of $36$ .

$\log (6 (6) \frac{x^{2} - 2}{6 x}) = 0$

Step 3.1.3.2

Factor $6$ out of $6 x$ .

$\log (6 (6) \frac{x^{2} - 2}{6 (x)}) = 0$

Step 3.1.3.3

Cancel the common factor.

$\log (6 \cdot 6 \frac{x^{2} - 2}{6 x}) = 0$

Step 3.1.3.4

Rewrite the expression.

$\log (6 \frac{x^{2} - 2}{x}) = 0$

Step 3.1.4

Combine $6$ and $\frac{x^{2} - 2}{x}$ .

$\log (\frac{6 (x^{2} - 2)}{x}) = 0$

Step 4

Rewrite $\log (\frac{6 (x^{2} - 2)}{x}) = 0$ 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 $b^{y} = x$ .

$10^{0} = \frac{6 (x^{2} - 2)}{x}$

Step 5

Cross multiply to remove the fraction.

$6 (x^{2} - 2) = 10^{0} (x)$

Step 6

Simplify $10^{0} (x)$ .

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Step 6.1

Anything raised to $0$ is $1$ .

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

Step 6.2

Multiply $x$ by $1$ .

$6 (x^{2} - 2) = x$

Step 7

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

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Step 7.1

Subtract $x$ from both sides of the equation.

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

Step 7.2

Simplify each term.

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Step 7.2.1

Apply the distributive property.

$6 x^{2} + 6 \cdot - 2 - x = 0$

Step 7.2.2

Multiply $6$ by $- 2$ .

$6 x^{2} - 12 - x = 0$

Step 8

Factor by grouping.

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Step 8.1

Reorder terms.

$6 x^{2} - x - 12 = 0$

Step 8.2

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 $a \cdot c = 6 \cdot - 12 = - 72$ and whose sum is $b = - 1$ .

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Step 8.2.1

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

$6 x^{2} - (x) - 12 = 0$

Step 8.2.2

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

$6 x^{2} + (8 - 9) x - 12 = 0$

Step 8.2.3

Apply the distributive property.

$6 x^{2} + 8 x - 9 x - 12 = 0$

Step 8.3

Factor out the greatest common factor from each group.

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Step 8.3.1

Group the first two terms and the last two terms.

$(6 x^{2} + 8 x) - 9 x - 12 = 0$

Step 8.3.2

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

$2 x (3 x + 4) - 3 (3 x + 4) = 0$

Step 8.4

Factor the polynomial by factoring out the greatest common factor, $3 x + 4$ .

$(3 x + 4) (2 x - 3) = 0$

Step 9

Simplify $(3 x + 4) (2 x - 3)$ .

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Step 9.1

Expand $(3 x + 4) (2 x - 3)$ using the FOIL Method.

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Step 9.1.1

Apply the distributive property.

$3 x (2 x - 3) + 4 (2 x - 3) = 0$

Step 9.1.2

Apply the distributive property.

$3 x (2 x) + 3 x \cdot - 3 + 4 (2 x - 3) = 0$

Step 9.1.3

Apply the distributive property.

$3 x (2 x) + 3 x \cdot - 3 + 4 (2 x) + 4 \cdot - 3 = 0$

Step 9.2

Simplify and combine like terms.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

Rewrite using the commutative property of multiplication.

$3 \cdot 2 x \cdot x + 3 x \cdot - 3 + 4 (2 x) + 4 \cdot - 3 = 0$

Step 9.2.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 9.2.1.2.1

Move $x$ .

$3 \cdot 2 (x \cdot x) + 3 x \cdot - 3 + 4 (2 x) + 4 \cdot - 3 = 0$

Step 9.2.1.2.2

Multiply $x$ by $x$ .

$3 \cdot 2 x^{2} + 3 x \cdot - 3 + 4 (2 x) + 4 \cdot - 3 = 0$

Step 9.2.1.3

Multiply $3$ by $2$ .

$6 x^{2} + 3 x \cdot - 3 + 4 (2 x) + 4 \cdot - 3 = 0$

Step 9.2.1.4

Multiply $- 3$ by $3$ .

$6 x^{2} - 9 x + 4 (2 x) + 4 \cdot - 3 = 0$

Step 9.2.1.5

Multiply $2$ by $4$ .

$6 x^{2} - 9 x + 8 x + 4 \cdot - 3 = 0$

Step 9.2.1.6

Multiply $4$ by $- 3$ .

$6 x^{2} - 9 x + 8 x - 12 = 0$

Step 9.2.2

Add $- 9 x$ and $8 x$ .

$6 x^{2} - x - 12 = 0$

Step 10

Factor by grouping.

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Step 10.1

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 $a \cdot c = 6 \cdot - 12 = - 72$ and whose sum is $b = - 1$ .

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Step 10.1.1

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

$6 x^{2} - (x) - 12 = 0$

Step 10.1.2

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

$6 x^{2} + (8 - 9) x - 12 = 0$

Step 10.1.3

Apply the distributive property.

$6 x^{2} + 8 x - 9 x - 12 = 0$

Step 10.2

Factor out the greatest common factor from each group.

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Step 10.2.1

Group the first two terms and the last two terms.

$(6 x^{2} + 8 x) - 9 x - 12 = 0$

Step 10.2.2

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

$2 x (3 x + 4) - 3 (3 x + 4) = 0$

Step 10.3

Factor the polynomial by factoring out the greatest common factor, $3 x + 4$ .

$(3 x + 4) (2 x - 3) = 0$

Step 11

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

$3 x + 4 = 0$

$2 x - 3 = 0$

Step 12

Set $3 x + 4$ equal to $0$ and solve for $x$ .

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Step 12.1

Set $3 x + 4$ equal to $0$ .

$3 x + 4 = 0$

Step 12.2

Solve $3 x + 4 = 0$ for $x$ .

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Step 12.2.1

Subtract $4$ from both sides of the equation.

$3 x = - 4$

Step 12.2.2

Divide each term in $3 x = - 4$ by $3$ and simplify.

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Step 12.2.2.1

Divide each term in $3 x = - 4$ by $3$ .

$\frac{3 x}{3} = \frac{- 4}{3}$

Step 12.2.2.2

Simplify the left side.

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Step 12.2.2.2.1

Cancel the common factor of $3$ .

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Step 12.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 4}{3}$

Step 12.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{3}$

Step 12.2.2.3

Simplify the right side.

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Step 12.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{4}{3}$

Step 13

Set $2 x - 3$ equal to $0$ and solve for $x$ .

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Step 13.1

Set $2 x - 3$ equal to $0$ .

$2 x - 3 = 0$

Step 13.2

Solve $2 x - 3 = 0$ for $x$ .

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Step 13.2.1

Add $3$ to both sides of the equation.

$2 x = 3$

Step 13.2.2

Divide each term in $2 x = 3$ by $2$ and simplify.

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Step 13.2.2.1

Divide each term in $2 x = 3$ by $2$ .

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

Step 13.2.2.2

Simplify the left side.

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Step 13.2.2.2.1

Cancel the common factor of $2$ .

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Step 13.2.2.2.1.1

Cancel the common factor.

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

Step 13.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 14

The final solution is all the values that make $(3 x + 4) (2 x - 3) = 0$ true.

$x = - \frac{4}{3}, \frac{3}{2}$

Step 15

Exclude the solutions that do not make $\log (x^{2} - 2) + 2 \log (6) = \log (6 x)$ true.

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

Step 16

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 1.5$

Mixed Number Form:

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