Finite Math Examples

$\log_{3} (x + 18) - \log_{3} (x - 8) = 3$

Step 1

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

$\log_{3} (\frac{x + 18}{x - 8}) = 3$

Step 2

Rewrite $\log_{3} (\frac{x + 18}{x - 8}) = 3$ 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 $b^{y} = x$ .

$3^{3} = \frac{x + 18}{x - 8}$

Step 3

Cross multiply to remove the fraction.

$x + 18 = 3^{3} (x - 8)$

Step 4

Simplify $3^{3} (x - 8)$ .

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Step 4.1

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

$x + 18 = 27 (x - 8)$

Step 4.2

Apply the distributive property.

$x + 18 = 27 x + 27 \cdot - 8$

Step 4.3

Multiply $27$ by $- 8$ .

$x + 18 = 27 x - 216$

Step 5

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

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Step 5.1

Subtract $27 x$ from both sides of the equation.

$x + 18 - 27 x = - 216$

Step 5.2

Subtract $27 x$ from $x$ .

$- 26 x + 18 = - 216$

Step 6

Factor $2$ out of $- 26 x + 18$ .

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Step 6.1

Factor $2$ out of $- 26 x$ .

$2 (- 13 x) + 18 = - 216$

Step 6.2

Factor $2$ out of $18$ .

$2 (- 13 x) + 2 (9) = - 216$

Step 6.3

Factor $2$ out of $2 (- 13 x) + 2 (9)$ .

$2 (- 13 x + 9) = - 216$

Step 7

Divide each term in $2 (- 13 x + 9) = - 216$ by $2$ and simplify.

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Step 7.1

Divide each term in $2 (- 13 x + 9) = - 216$ by $2$ .

$\frac{2 (- 13 x + 9)}{2} = \frac{- 216}{2}$

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.

$\frac{2 (- 13 x + 9)}{2} = \frac{- 216}{2}$

Step 7.2.1.2

Divide $- 13 x + 9$ by $1$ .

$- 13 x + 9 = \frac{- 216}{2}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $- 216$ by $2$ .

$- 13 x + 9 = - 108$

Step 8

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

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Step 8.1

Subtract $9$ from both sides of the equation.

$- 13 x = - 108 - 9$

Step 8.2

Subtract $9$ from $- 108$ .

$- 13 x = - 117$

Step 9

Divide each term in $- 13 x = - 117$ by $- 13$ and simplify.

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Step 9.1

Divide each term in $- 13 x = - 117$ by $- 13$ .

$\frac{- 13 x}{- 13} = \frac{- 117}{- 13}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $- 13$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{- 13 x}{- 13} = \frac{- 117}{- 13}$

Step 9.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 117}{- 13}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Divide $- 117$ by $- 13$ .

$x = 9$