Finite Math Examples

$\ln ({(\frac{3 x - 2}{3 x + 1})}^{2 x - 1})$

Step 1

Set the argument in $\ln ({(\frac{3 x - 2}{3 x + 1})}^{2 x - 1})$ greater than $0$ to find where the expression is defined.

${(\frac{3 x - 2}{3 x + 1})}^{2 x - 1} > 0$

Step 2

Solve for $x$ .

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Step 2.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln ({(\frac{3 x - 2}{3 x + 1})}^{2 x - 1}) > \ln (0)$

Step 2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 2.3

There is no solution for ${(\frac{3 x - 2}{3 x + 1})}^{2 x - 1} > 0$

No solution

Step 3

Set the denominator in $\frac{3 x - 2}{3 x + 1}$ equal to $0$ to find where the expression is undefined.

$3 x + 1 = 0$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Step 4.2

Divide each term in $3 x = - 1$ by $3$ and simplify.

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Step 4.2.1

Divide each term in $3 x = - 1$ by $3$ .

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{3}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

Step 5

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

Interval Notation:

$(- \infty, - \frac{1}{3}) \cup (- \frac{1}{3}, \infty)$

Set-Builder Notation:

${x | x \neq - \frac{1}{3}}$

Step 6