Algebra Examples

$\ln (3 x - 1) - \ln (x + 2) = 0$

Step 1

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

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

Step 2

Rewrite $\ln (\frac{3 x - 1}{x + 2}) = 0$ 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 $b^{y} = x$ .

$e^{0} = \frac{3 x - 1}{x + 2}$

Step 3

Cross multiply to remove the fraction.

$3 x - 1 = e^{0} (x + 2)$

Step 4

Simplify $e^{0} (x + 2)$ .

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Step 4.1

Anything raised to $0$ is $1$ .

$3 x - 1 = 1 (x + 2)$

Step 4.2

Multiply $x + 2$ by $1$ .

$3 x - 1 = x + 2$

Step 5

Move all terms containing $x$ to the left side of the equation.

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Step 5.1

Subtract $x$ from both sides of the equation.

$3 x - 1 - x = 2$

Step 5.2

Subtract $x$ from $3 x$ .

$2 x - 1 = 2$

Step 6

Move all terms not containing $x$ to the right side of the equation.

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Step 6.1

Add $1$ to both sides of the equation.

$2 x = 2 + 1$

Step 6.2

Add $2$ and $1$ .

$2 x = 3$

Step 7

Divide each term in $2 x = 3$ by $2$ and simplify.

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Step 7.1

Divide each term in $2 x = 3$ by $2$ .

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

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $2$ .

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Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Divide $x$ by $1$ .

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 1.5$

Mixed Number Form:

$x = 1 \frac{1}{2}$