Linear Algebra Examples

$\log_{3} (5 x + 2) - \log_{3} (x - 2) = 2$

Step 1

Set the argument in $\log_{3} (5 x + 2)$ greater than $0$ to find where the expression is defined.

$5 x + 2 > 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $2$ from both sides of the inequality.

$5 x > - 2$

Step 2.2

Divide each term in $5 x > - 2$ by $5$ and simplify.

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Step 2.2.1

Divide each term in $5 x > - 2$ by $5$ .

$\frac{5 x}{5} > \frac{- 2}{5}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $5$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} > \frac{- 2}{5}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{- 2}{5}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Move the negative in front of the fraction.

$x > - \frac{2}{5}$

Step 3

Set the argument in $\log_{3} (x - 2)$ greater than $0$ to find where the expression is defined.

$x - 2 > 0$

Step 4

Add $2$ to both sides of the inequality.

$x > 2$

Step 5

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

Interval Notation:

$(2, \infty)$

Set-Builder Notation:

${x | x > 2}$

Step 6