Solve the Inequality for x log base 3 of x-3 log base 3 of 2<=1

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

Convert the inequality to an equality.

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

Solve the equation.

Simplify the left side.

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

Simplify each term.

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

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

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

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

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

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

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 $b^{y} = x$ .

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

Solve for $x$ .

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

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

Multiply both sides of the equation by $8$ .

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

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the right side.

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

Evaluate the exponent.

$x = 8 \cdot 3$

Multiply $8$ by $3$ .

$x = 24$

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

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

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

Solve for $x$ .

Multiply both sides by $8$ .

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

Simplify.

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

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

Rewrite the expression.

$x > 0 \cdot 8$

Simplify the right side.

Multiply $0$ by $8$ .

$x > 0$

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

$(0, \infty)$

Use each root to create test intervals.

$x < 0$

$0 < x < 24$

$x > 24$

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 $x < 0$ 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.

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

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

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

$x = 2$

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

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

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

True

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

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

$x = 26$

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

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

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

False

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

The solution consists of all of the true intervals.

$0 < x \leq 24$

The result can be shown in multiple forms.

Inequality Form:

$0 < x \leq 24$

Interval Notation:

$(0, 24]$

Solve the Inequality for x log base 3 of x-3 log base 3 of 2<=1

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