Precalculus Examples

$\log_{0.5} (x^{2} + x) - \log_{0.5} (x^{2} - x) = - 2$

Step 1

Simplify the left side.

Tap for more steps...

Step 1.1

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

$\log_{0.5} (\frac{x^{2} + x}{x^{2} - x}) = - 2$

Step 1.2

Factor $x$ out of $x^{2} + x$ .

Tap for more steps...

Step 1.2.1

Factor $x$ out of $x^{2}$ .

$\log_{0.5} (\frac{x \cdot x + x}{x^{2} - x}) = - 2$

Step 1.2.2

Raise $x$ to the power of $1$ .

$\log_{0.5} (\frac{x \cdot x + x}{x^{2} - x}) = - 2$

Step 1.2.3

Factor $x$ out of $x^{1}$ .

$\log_{0.5} (\frac{x \cdot x + x \cdot 1}{x^{2} - x}) = - 2$

Step 1.2.4

Factor $x$ out of $x \cdot x + x \cdot 1$ .

$\log_{0.5} (\frac{x (x + 1)}{x^{2} - x}) = - 2$

Step 1.3

Factor $x$ out of $x^{2} - x$ .

Tap for more steps...

Step 1.3.1

Factor $x$ out of $x^{2}$ .

$\log_{0.5} (\frac{x (x + 1)}{x \cdot x - x}) = - 2$

Step 1.3.2

Factor $x$ out of $- x$ .

$\log_{0.5} (\frac{x (x + 1)}{x \cdot x + x \cdot - 1}) = - 2$

Step 1.3.3

Factor $x$ out of $x \cdot x + x \cdot - 1$ .

$\log_{0.5} (\frac{x (x + 1)}{x (x - 1)}) = - 2$

Step 1.4

Cancel the common factor of $x$ .

Tap for more steps...

Step 1.4.1

Cancel the common factor.

$\log_{0.5} (\frac{x (x + 1)}{x (x - 1)}) = - 2$

Step 1.4.2

Rewrite the expression.

$\log_{0.5} (\frac{x + 1}{x - 1}) = - 2$

Step 2

Rewrite $\log_{0.5} (\frac{x + 1}{x - 1}) = - 2$ 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 $b^{y} = x$ .

${0.5}^{- 2} = \frac{x + 1}{x - 1}$

Step 3

Cross multiply to remove the fraction.

$x + 1 = {0.5}^{- 2} (x - 1)$

Step 4

Simplify ${0.5}^{- 2} (x - 1)$ .

Tap for more steps...

Step 4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x + 1 = \frac{1}{{0.5}^{2}} (x - 1)$

Step 4.2

Simplify the expression.

Tap for more steps...

Step 4.2.1

Raise $0.5$ to the power of $2$ .

$x + 1 = \frac{1}{0.25} (x - 1)$

Step 4.2.2

Divide $1$ by $0.25$ .

$x + 1 = 4 (x - 1)$

Step 4.3

Apply the distributive property.

$x + 1 = 4 x + 4 \cdot - 1$

Step 4.4

Multiply $4$ by $- 1$ .

$x + 1 = 4 x - 4$

Step 5

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

Tap for more steps...

Step 5.1

Subtract $4 x$ from both sides of the equation.

$x + 1 - 4 x = - 4$

Step 5.2

Subtract $4 x$ from $x$ .

$- 3 x + 1 = - 4$

Step 6

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

Tap for more steps...

Step 6.1

Subtract $1$ from both sides of the equation.

$- 3 x = - 4 - 1$

Step 6.2

Subtract $1$ from $- 4$ .

$- 3 x = - 5$

Step 7

Divide each term in $- 3 x = - 5$ by $- 3$ and simplify.

Tap for more steps...

Step 7.1

Divide each term in $- 3 x = - 5$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 5}{- 3}$

Step 7.2

Simplify the left side.

Tap for more steps...

Step 7.2.1

Cancel the common factor of $- 3$ .

Tap for more steps...

Step 7.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 5}{- 3}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{- 3}$

Step 7.3

Simplify the right side.

Tap for more steps...

Step 7.3.1

Dividing two negative values results in a positive value.

$x = \frac{5}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{3}$

Decimal Form:

$x = 1.\overline{6}$

Mixed Number Form:

$x = 1 \frac{2}{3}$