Finite Math Examples

$\log_{3} (x + 4) - \log_{3} (x - 4) = \log_{3} (5)$

Step 1

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

$\log_{3} (\frac{x + 4}{x - 4}) = \log_{3} (5)$

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$\frac{x + 4}{x - 4} = 5$

Step 3

Solve for $x$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

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

$x - 4, 1$

Step 3.1.2

Remove parentheses.

$x - 4, 1$

Step 3.1.3

The LCM of one and any expression is the expression.

$x - 4$

Step 3.2

Multiply each term in $\frac{x + 4}{x - 4} = 5$ by $x - 4$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{x + 4}{x - 4} = 5$ by $x - 4$ .

$\frac{x + 4}{x - 4} (x - 4) = 5 (x - 4)$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $x - 4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{x + 4}{x - 4} (x - 4) = 5 (x - 4)$

Step 3.2.2.1.2

Rewrite the expression.

$x + 4 = 5 (x - 4)$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Apply the distributive property.

$x + 4 = 5 x + 5 \cdot - 4$

Step 3.2.3.2

Multiply $5$ by $- 4$ .

$x + 4 = 5 x - 20$

Step 3.3

Solve the equation.

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Step 3.3.1

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

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Step 3.3.1.1

Subtract $5 x$ from both sides of the equation.

$x + 4 - 5 x = - 20$

Step 3.3.1.2

Subtract $5 x$ from $x$ .

$- 4 x + 4 = - 20$

Step 3.3.2

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

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Step 3.3.2.1

Subtract $4$ from both sides of the equation.

$- 4 x = - 20 - 4$

Step 3.3.2.2

Subtract $4$ from $- 20$ .

$- 4 x = - 24$

Step 3.3.3

Divide each term in $- 4 x = - 24$ by $- 4$ and simplify.

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Step 3.3.3.1

Divide each term in $- 4 x = - 24$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 24}{- 4}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 4$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 24}{- 4}$

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 24}{- 4}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Divide $- 24$ by $- 4$ .

$x = 6$