Algebra Examples

$\log_{4} (x) + \log_{4} (x + 6) = 2$

Step 1

Subtract $2$ from both sides of the equation.

$\log_{4} (x) + \log_{4} (x + 6) - 2 = 0$

Step 2

Simplify $\log_{4} (x) + \log_{4} (x + 6) - 2$ .

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Step 2.1

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

$\log_{4} (x (x + 6)) - 2 = 0$

Step 2.2

Simplify each term.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Multiply $x$ by $x$ .

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

Step 2.2.3

Move $6$ to the left of $x$ .

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

Step 3

Set the argument in $\log_{4} (x^{2} + 6 x)$ less than or equal to $0$ to find where the expression is undefined.

$x^{2} + 6 x \leq 0$

Step 4

Solve for $x$ .

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Step 4.1

Convert the inequality to an equation.

$x^{2} + 6 x = 0$

Step 4.2

Factor $x$ out of $x^{2} + 6 x$ .

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Step 4.2.1

Factor $x$ out of $x^{2}$ .

$x \cdot x + 6 x = 0$

Step 4.2.2

Factor $x$ out of $6 x$ .

$x \cdot x + x \cdot 6 = 0$

Step 4.2.3

Factor $x$ out of $x \cdot x + x \cdot 6$ .

$x (x + 6) = 0$

Step 4.3

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

$x = 0$

$x + 6 = 0$

Step 4.4

Set $x$ equal to $0$ .

$x = 0$

Step 4.5

Set $x + 6$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Step 4.5.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 4.6

The final solution is all the values that make $x (x + 6) = 0$ true.

$x = 0, - 6$

Step 4.7

Use each root to create test intervals.

$x < - 6$

$- 6 < x < 0$

$x > 0$

Step 4.8

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 4.8.1

Test a value on the interval $x < - 6$ to see if it makes the inequality true.

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Step 4.8.1.1

Choose a value on the interval $x < - 6$ and see if this value makes the original inequality true.

$x = - 8$

Step 4.8.1.2

Replace $x$ with $- 8$ in the original inequality.

${(- 8)}^{2} + 6 (- 8) \leq 0$

Step 4.8.1.3

The left side $16$ is greater than the right side $0$ , which means that the given statement is false.

False

Step 4.8.2

Test a value on the interval $- 6 < x < 0$ to see if it makes the inequality true.

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Step 4.8.2.1

Choose a value on the interval $- 6 < x < 0$ and see if this value makes the original inequality true.

$x = - 3$

Step 4.8.2.2

Replace $x$ with $- 3$ in the original inequality.

${(- 3)}^{2} + 6 (- 3) \leq 0$

Step 4.8.2.3

The left side $- 9$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 4.8.3

Test a value on the interval $x > 0$ to see if it makes the inequality true.

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Step 4.8.3.1

Choose a value on the interval $x > 0$ and see if this value makes the original inequality true.

$x = 2$

Step 4.8.3.2

Replace $x$ with $2$ in the original inequality.

${(2)}^{2} + 6 (2) \leq 0$

Step 4.8.3.3

The left side $16$ is greater than the right side $0$ , which means that the given statement is false.

False

Step 4.8.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 6$ False

$- 6 < x < 0$ True

$x > 0$ False

$x < - 6$ False

$- 6 < x < 0$ True

$x > 0$ False

Step 4.9

The solution consists of all of the true intervals.

$- 6 \leq x \leq 0$

Step 5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$- 6 \leq x \leq 0$

$[- 6, 0]$

Step 6