Algebra Examples

$\log_{3} (x) - 3 \log_{3} (2) \leq 1$

Step 1

Convert the inequality to an equality.

$\log_{3} (x) - 3 \log_{3} (2) = 1$

Step 2

Solve the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\log_{3} (x) - 3 \log_{3} (2)$ .

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Step 2.1.1.1

Simplify each term.

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Step 2.1.1.1.1

Simplify $- 3 \log_{3} (2)$ by moving $3$ inside the logarithm.

$\log_{3} (x) - \log_{3} (2^{3}) = 1$

Step 2.1.1.1.2

Raise $2$ to the power of $3$ .

$\log_{3} (x) - \log_{3} (8) = 1$

Step 2.1.1.2

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

$\log_{3} (\frac{x}{8}) = 1$

Step 2.2

Rewrite $\log_{3} (\frac{x}{8}) = 1$ 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 $b^{y} = x$ .

$3^{1} = \frac{x}{8}$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Rewrite the equation as $\frac{x}{8} = 3^{1}$ .

$\frac{x}{8} = 3$

Step 2.3.2

Multiply both sides of the equation by $8$ .

$8 \frac{x}{8} = 8 \cdot 3^{1}$

Step 2.3.3

Simplify both sides of the equation.

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Step 2.3.3.1

Simplify the left side.

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Step 2.3.3.1.1

Cancel the common factor of $8$ .

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Step 2.3.3.1.1.1

Cancel the common factor.

$8 \frac{x}{8} = 8 \cdot 3^{1}$

Step 2.3.3.1.1.2

Rewrite the expression.

$x = 8 \cdot 3^{1}$

Step 2.3.3.2

Simplify the right side.

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Step 2.3.3.2.1

Simplify $8 \cdot 3^{1}$ .

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Step 2.3.3.2.1.1

Evaluate the exponent.

$x = 8 \cdot 3$

Step 2.3.3.2.1.2

Multiply $8$ by $3$ .

$x = 24$

Step 3

Find the domain of $\log_{3} (\frac{x}{8}) - 1$ .

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Step 3.1

Set the argument in $\log_{3} (\frac{x}{8})$ greater than $0$ to find where the expression is defined.

$\frac{x}{8} > 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Multiply both sides by $8$ .

$\frac{x}{8} \cdot 8 > 0 \cdot 8$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Simplify the left side.

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Step 3.2.2.1.1

Cancel the common factor of $8$ .

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Step 3.2.2.1.1.1

Cancel the common factor.

$\frac{x}{8} \cdot 8 > 0 \cdot 8$

Step 3.2.2.1.1.2

Rewrite the expression.

$x > 0 \cdot 8$

Step 3.2.2.2

Simplify the right side.

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Step 3.2.2.2.1

Multiply $0$ by $8$ .

$x > 0$

Step 3.3

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

$(0, \infty)$

Step 4

Use each root to create test intervals.

$x < 0$

$0 < x < 24$

$x > 24$

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 $x < 0$ to see if it makes the inequality true.

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Step 5.1.1

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

$x = - 2$

Step 5.1.2

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

$\log_{3} (- 2) - 3 \log_{3} (2) \leq 1$

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 $0 < x < 24$ to see if it makes the inequality true.

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Step 5.2.1

Choose a value on the interval $0 < x < 24$ and see if this value makes the original inequality true.

$x = 2$

Step 5.2.2

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

$\log_{3} (2) - 3 \log_{3} (2) \leq 1$

Step 5.2.3

The left side $- 1.2618595$ is less than the right side $1$ , which means that the given statement is always true.

True

Step 5.3

Test a value on the interval $x > 24$ to see if it makes the inequality true.

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Step 5.3.1

Choose a value on the interval $x > 24$ and see if this value makes the original inequality true.

$x = 26$

Step 5.3.2

Replace $x$ with $26$ in the original inequality.

$\log_{3} (26) - 3 \log_{3} (2) \leq 1$

Step 5.3.3

The left side $1.07285801$ is greater than the right side $1$ , which means that the given statement is false.

False

Step 5.4

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

$x < 0$ False

$0 < x < 24$ True

$x > 24$ False

$x < 0$ False

$0 < x < 24$ True

$x > 24$ False

Step 6

The solution consists of all of the true intervals.

$0 < x \leq 24$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$0 < x \leq 24$

Interval Notation:

$(0, 24]$

Step 8