Algebra Examples

$\log_{3} (x + 9) < 4$

Step 1

Convert the inequality to an equality.

$\log_{3} (x + 9) = 4$

Step 2

Solve the equation.

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Step 2.1

Rewrite $\log_{3} (x + 9) = 4$ 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 $b^{y} = x$ .

$3^{4} = x + 9$

Step 2.2

Solve for $x$ .

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Step 2.2.1

Rewrite the equation as $x + 9 = 3^{4}$ .

$x + 9 = 3^{4}$

Step 2.2.2

Raise $3$ to the power of $4$ .

$x + 9 = 81$

Step 2.2.3

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

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Step 2.2.3.1

Subtract $9$ from both sides of the equation.

$x = 81 - 9$

Step 2.2.3.2

Subtract $9$ from $81$ .

$x = 72$

Step 3

Find the domain of $\log_{3} (x + 9) - 4$ .

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Step 3.1

Set the argument in $\log_{3} (x + 9)$ greater than $0$ to find where the expression is defined.

$x + 9 > 0$

Step 3.2

Subtract $9$ from both sides of the inequality.

$x > - 9$

Step 3.3

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

$(- 9, \infty)$

Step 4

Use each root to create test intervals.

$x < - 9$

$- 9 < x < 72$

$x > 72$

Step 5

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 5.1

Test a value on the interval $x < - 9$ to see if it makes the inequality true.

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Step 5.1.1

Choose a value on the interval $x < - 9$ and see if this value makes the original inequality true.

$x = - 12$

Step 5.1.2

Replace $x$ with $- 12$ in the original inequality.

$\log_{3} ((- 12) + 9) < 4$

Step 5.1.3

Determine if the inequality is true.

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Step 5.1.3.1

The equation cannot be solved because it is undefined.

$Undefined$

Step 5.1.3.2

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

False

Step 5.2

Test a value on the interval $- 9 < x < 72$ to see if it makes the inequality true.

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Step 5.2.1

Choose a value on the interval $- 9 < x < 72$ and see if this value makes the original inequality true.

$x = 0$

Step 5.2.2

Replace $x$ with $0$ in the original inequality.

$\log_{3} ((0) + 9) < 4$

Step 5.2.3

The left side $2$ is less than the right side $4$ , which means that the given statement is always true.

True

Step 5.3

Test a value on the interval $x > 72$ to see if it makes the inequality true.

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Step 5.3.1

Choose a value on the interval $x > 72$ and see if this value makes the original inequality true.

$x = 74$

Step 5.3.2

Replace $x$ with $74$ in the original inequality.

$\log_{3} ((74) + 9) < 4$

Step 5.3.3

The left side $4.02220205$ is not less than the right side $4$ , which means that the given statement is false.

False

Step 5.4

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

$x < - 9$ False

$- 9 < x < 72$ True

$x > 72$ False

$x < - 9$ False

$- 9 < x < 72$ True

$x > 72$ False

Step 6

The solution consists of all of the true intervals.

$- 9 < x < 72$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$- 9 < x < 72$

Interval Notation:

$(- 9, 72)$

Step 8