Algebra Examples

$\log_{3} (x) + \log_{3} (6) \geq 2$

Step 1

Convert the inequality to an equality.

$\log_{3} (x) + \log_{3} (6) = 2$

Step 2

Solve the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

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

$\log_{3} (x \cdot 6) = 2$

Step 2.1.2

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

$\log_{3} (6 x) = 2$

Step 2.2

Rewrite $\log_{3} (6 x) = 2$ 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 $b^{y} = x$ .

$3^{2} = 6 x$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Rewrite the equation as $6 x = 3^{2}$ .

$6 x = 3^{2}$

Step 2.3.2

Divide each term in $6 x = 3^{2}$ by $6$ and simplify.

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Step 2.3.2.1

Divide each term in $6 x = 3^{2}$ by $6$ .

$\frac{6 x}{6} = \frac{3^{2}}{6}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $6$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{3^{2}}{6}$

Step 2.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3^{2}}{6}$

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

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

$x = \frac{9}{6}$

Step 2.3.2.3.2

Cancel the common factor of $9$ and $6$ .

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Step 2.3.2.3.2.1

Factor $3$ out of $9$ .

$x = \frac{3 (3)}{6}$

Step 2.3.2.3.2.2

Cancel the common factors.

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Step 2.3.2.3.2.2.1

Factor $3$ out of $6$ .

$x = \frac{3 \cdot 3}{3 \cdot 2}$

Step 2.3.2.3.2.2.2

Cancel the common factor.

$x = \frac{3 \cdot 3}{3 \cdot 2}$

Step 2.3.2.3.2.2.3

Rewrite the expression.

$x = \frac{3}{2}$

Step 3

Find the domain of $\log_{3} (6 x) - 2$ .

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Step 3.1

Set the argument in $\log_{3} (6 x)$ greater than $0$ to find where the expression is defined.

$6 x > 0$

Step 3.2

Divide each term in $6 x > 0$ by $6$ and simplify.

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Step 3.2.1

Divide each term in $6 x > 0$ by $6$ .

$\frac{6 x}{6} > \frac{0}{6}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $6$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} > \frac{0}{6}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{0}{6}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $0$ by $6$ .

$x > 0$

Step 3.3

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

$(0, \infty)$

Step 4

The solution consists of all of the true intervals.

$x \geq \frac{3}{2}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x \geq \frac{3}{2}$

Interval Notation:

$[\frac{3}{2}, \infty)$

Step 6