Finite Math Examples

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

Step 1

Subtract $3$ from both sides of the equation.

$\log_{2} (x) - \log_{2} (x - 2) - 3 = 0$

Step 2

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

$\log_{2} (\frac{x}{x - 2}) - 3 = 0$

Step 3

Add $3$ to both sides of the equation.

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

Step 4

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

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

Step 5

Solve for $x$ .

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Step 5.1

Rewrite the equation as $\frac{x}{x - 2} = 2^{3}$ .

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

Step 5.2

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

$\frac{x}{x - 2} = 8$

Step 5.3

Find the LCD of the terms in the equation.

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Step 5.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x - 2, 1$

Step 5.3.2

Remove parentheses.

$x - 2, 1$

Step 5.3.3

The LCM of one and any expression is the expression.

$x - 2$

Step 5.4

Multiply each term in $\frac{x}{x - 2} = 8$ by $x - 2$ to eliminate the fractions.

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Step 5.4.1

Multiply each term in $\frac{x}{x - 2} = 8$ by $x - 2$ .

$\frac{x}{x - 2} (x - 2) = 8 (x - 2)$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $x - 2$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{x}{x - 2} (x - 2) = 8 (x - 2)$

Step 5.4.2.1.2

Rewrite the expression.

$x = 8 (x - 2)$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Apply the distributive property.

$x = 8 x + 8 \cdot - 2$

Step 5.4.3.2

Multiply $8$ by $- 2$ .

$x = 8 x - 16$

Step 5.5

Solve the equation.

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Step 5.5.1

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

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Step 5.5.1.1

Subtract $8 x$ from both sides of the equation.

$x - 8 x = - 16$

Step 5.5.1.2

Subtract $8 x$ from $x$ .

$- 7 x = - 16$

Step 5.5.2

Divide each term in $- 7 x = - 16$ by $- 7$ and simplify.

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Step 5.5.2.1

Divide each term in $- 7 x = - 16$ by $- 7$ .

$\frac{- 7 x}{- 7} = \frac{- 16}{- 7}$

Step 5.5.2.2

Simplify the left side.

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Step 5.5.2.2.1

Cancel the common factor of $- 7$ .

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Step 5.5.2.2.1.1

Cancel the common factor.

$\frac{- 7 x}{- 7} = \frac{- 16}{- 7}$

Step 5.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 16}{- 7}$

Step 5.5.2.3

Simplify the right side.

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Step 5.5.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{16}{7}$