Solve for x log base 2 of x+ log base 2 of x+4<5

$\log_{2} (x) + \log_{2} (x + 4) < 5$

Convert the inequality to an equality.

$\log_{2} (x) + \log_{2} (x + 4) = 5$

Solve the equation.

Simplify the left side.

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

$\log_{2} (x (x + 4)) = 5$

Apply the distributive property.

$\log_{2} (x \cdot x + x \cdot 4) = 5$

Simplify the expression.

Multiply $x$ by $x$ .

$\log_{2} (x^{2} + x \cdot 4) = 5$

Move $4$ to the left of $x$ .

$\log_{2} (x^{2} + 4 x) = 5$

Rewrite $\log_{2} (x^{2} + 4 x) = 5$ 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 $b^{y} = x$ .

$2^{5} = x^{2} + 4 x$

Solve for $x$ .

Rewrite the equation as $x^{2} + 4 x = 2^{5}$ .

$x^{2} + 4 x = 2^{5}$

Raise $2$ to the power of $5$ .

$x^{2} + 4 x = 32$

Subtract $32$ from both sides of the equation.

$x^{2} + 4 x - 32 = 0$

Factor $x^{2} + 4 x - 32$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 32$ and whose sum is $4$ .

$- 4, 8$

Write the factored form using these integers.

$(x - 4) (x + 8) = 0$

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

$x - 4 = 0$

$x + 8 = 0$

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

Set $x + 8$ equal to $0$ and solve for $x$ .

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Subtract $8$ from both sides of the equation.

$x = - 8$

The final solution is all the values that make $(x - 4) (x + 8) = 0$ true.

$x = 4, - 8$

Find the domain of $\log_{2} (x^{2} + 4 x) - 5$ .

Set the argument in $\log_{2} (x^{2} + 4 x)$ greater than $0$ to find where the expression is defined.

$x^{2} + 4 x > 0$

Solve for $x$ .

Convert the inequality to an equation.

$x^{2} + 4 x = 0$

Factor $x$ out of $x^{2} + 4 x$ .

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

$x \cdot x + 4 x = 0$

Factor $x$ out of $4 x$ .

$x \cdot x + x \cdot 4 = 0$

Factor $x$ out of $x \cdot x + x \cdot 4$ .

$x (x + 4) = 0$

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

$x = 0$

$x + 4 = 0$

Set $x$ equal to $0$ .

$x = 0$

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

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

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

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

$x = 0, - 4$

Use each root to create test intervals.

$x < - 4$

$- 4 < x < 0$

$x > 0$

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

Test a value on the interval $x < - 4$ to see if it makes the inequality true.

Choose a value on the interval $x < - 4$ and see if this value makes the original inequality true.

$x = - 6$

Replace $x$ with $- 6$ in the original inequality.

${(- 6)}^{2} + 4 (- 6) > 0$

The left side $12$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- 4 < x < 0$ to see if it makes the inequality true.

Choose a value on the interval $- 4 < x < 0$ and see if this value makes the original inequality true.

$x = - 2$

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

${(- 2)}^{2} + 4 (- 2) > 0$

The left side $- 4$ is not greater than the right side $0$ , which means that the given statement is false.

False

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

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

$x = 2$

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

${(2)}^{2} + 4 (2) > 0$

The left side $12$ is greater than the right side $0$ , which means that the given statement is always true.

True

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

$x < - 4$ True

$- 4 < x < 0$ False

$x > 0$ True

$x < - 4$ True

$- 4 < x < 0$ False

$x > 0$ True

The solution consists of all of the true intervals.

$x < - 4$ or $x > 0$

The domain is all values of $x$ that make the expression defined.

$(- \infty, - 4) \cup (0, \infty)$

Use each root to create test intervals.

$x < - 8$

$- 8 < x < - 4$

$- 4 < x < 0$

$0 < x < 4$

$x > 4$

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

Test a value on the interval $x < - 8$ to see if it makes the inequality true.

Choose a value on the interval $x < - 8$ and see if this value makes the original inequality true.

$x = - 10$

Replace $x$ with $- 10$ in the original inequality.

$\log_{2} (- 10) + \log_{2} ((- 10) + 4) < 5$

Determine if the inequality is true.

The equation cannot be solved because it is undefined.

$Undefined$

The left side has no solution, which means that the given statement is false.

False

Test a value on the interval $- 8 < x < - 4$ to see if it makes the inequality true.

Choose a value on the interval $- 8 < x < - 4$ and see if this value makes the original inequality true.

$x = - 6$

Replace $x$ with $- 6$ in the original inequality.

$\log_{2} (- 6) + \log_{2} ((- 6) + 4) < 5$

Determine if the inequality is true.

The equation cannot be solved because it is undefined.

$Undefined$

The left side has no solution, which means that the given statement is false.

False

Test a value on the interval $- 4 < x < 0$ to see if it makes the inequality true.

Choose a value on the interval $- 4 < x < 0$ and see if this value makes the original inequality true.

$x = - 2$

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

$\log_{2} (- 2) + \log_{2} ((- 2) + 4) < 5$

Determine if the inequality is true.

The equation cannot be solved because it is undefined.

$Undefined$

The left side has no solution, which means that the given statement is false.

False

Test a value on the interval $0 < x < 4$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < 4$ and see if this value makes the original inequality true.

$x = 2$

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

$\log_{2} (2) + \log_{2} ((2) + 4) < 5$

The left side $3.5849625$ is less than the right side $5$ , which means that the given statement is always true.

True

Test a value on the interval $x > 4$ to see if it makes the inequality true.

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

Replace $x$ with $6$ in the original inequality.

$\log_{2} (6) + \log_{2} ((6) + 4) < 5$

The left side $5.90689059$ is not less than the right side $5$ , which means that the given statement is false.

False

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

$x < - 8$ False

$- 8 < x < - 4$ False

$- 4 < x < 0$ False

$0 < x < 4$ True

$x > 4$ False

$x < - 8$ False

$- 8 < x < - 4$ False

$- 4 < x < 0$ False

$0 < x < 4$ True

$x > 4$ False

The solution consists of all of the true intervals.

$0 < x < 4$

The result can be shown in multiple forms.

Inequality Form:

$0 < x < 4$

Interval Notation:

$(0, 4)$

Solve for x log base 2 of x+ log base 2 of x+4<5

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