Precalculus Examples

$\log_{5} (x + 9) - \log_{5} (x - 1) = 2$

Step 1

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

$\log_{5} (\frac{x + 9}{x - 1}) = 2$

Step 2

Rewrite $\log_{5} (\frac{x + 9}{x - 1}) = 2$ 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 $b^{y} = x$ .

$5^{2} = \frac{x + 9}{x - 1}$

Step 3

Cross multiply to remove the fraction.

$x + 9 = 5^{2} (x - 1)$

Step 4

Simplify $5^{2} (x - 1)$ .

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Step 4.1

Raise $5$ to the power of $2$ .

$x + 9 = 25 (x - 1)$

Step 4.2

Apply the distributive property.

$x + 9 = 25 x + 25 \cdot - 1$

Step 4.3

Multiply $25$ by $- 1$ .

$x + 9 = 25 x - 25$

Step 5

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

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Step 5.1

Subtract $25 x$ from both sides of the equation.

$x + 9 - 25 x = - 25$

Step 5.2

Subtract $25 x$ from $x$ .

$- 24 x + 9 = - 25$

Step 6

Factor $3$ out of $- 24 x + 9$ .

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Step 6.1

Factor $3$ out of $- 24 x$ .

$3 (- 8 x) + 9 = - 25$

Step 6.2

Factor $3$ out of $9$ .

$3 (- 8 x) + 3 (3) = - 25$

Step 6.3

Factor $3$ out of $3 (- 8 x) + 3 (3)$ .

$3 (- 8 x + 3) = - 25$

Step 7

Divide each term in $3 (- 8 x + 3) = - 25$ by $3$ and simplify.

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Step 7.1

Divide each term in $3 (- 8 x + 3) = - 25$ by $3$ .

$\frac{3 (- 8 x + 3)}{3} = \frac{- 25}{3}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $3$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{3 (- 8 x + 3)}{3} = \frac{- 25}{3}$

Step 7.2.1.2

Divide $- 8 x + 3$ by $1$ .

$- 8 x + 3 = \frac{- 25}{3}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Move the negative in front of the fraction.

$- 8 x + 3 = - \frac{25}{3}$

Step 8

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

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Step 8.1

Subtract $3$ from both sides of the equation.

$- 8 x = - \frac{25}{3} - 3$

Step 8.2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 8 x = - \frac{25}{3} - 3 \cdot \frac{3}{3}$

Step 8.3

Combine $- 3$ and $\frac{3}{3}$ .

$- 8 x = - \frac{25}{3} + \frac{- 3 \cdot 3}{3}$

Step 8.4

Combine the numerators over the common denominator.

$- 8 x = \frac{- 25 - 3 \cdot 3}{3}$

Step 8.5

Simplify the numerator.

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Step 8.5.1

Multiply $- 3$ by $3$ .

$- 8 x = \frac{- 25 - 9}{3}$

Step 8.5.2

Subtract $9$ from $- 25$ .

$- 8 x = \frac{- 34}{3}$

Step 8.6

Move the negative in front of the fraction.

$- 8 x = - \frac{34}{3}$

Step 9

Divide each term in $- 8 x = - \frac{34}{3}$ by $- 8$ and simplify.

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Step 9.1

Divide each term in $- 8 x = - \frac{34}{3}$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{- \frac{34}{3}}{- 8}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $- 8$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} = \frac{- \frac{34}{3}}{- 8}$

Step 9.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{34}{3}}{- 8}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{34}{3} \cdot \frac{1}{- 8}$

Step 9.3.2

Cancel the common factor of $2$ .

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Step 9.3.2.1

Move the leading negative in $- \frac{34}{3}$ into the numerator.

$x = \frac{- 34}{3} \cdot \frac{1}{- 8}$

Step 9.3.2.2

Factor $2$ out of $- 34$ .

$x = \frac{2 (- 17)}{3} \cdot \frac{1}{- 8}$

Step 9.3.2.3

Factor $2$ out of $- 8$ .

$x = \frac{2 \cdot - 17}{3} \cdot \frac{1}{2 \cdot - 4}$

Step 9.3.2.4

Cancel the common factor.

$x = \frac{2 \cdot - 17}{3} \cdot \frac{1}{2 \cdot - 4}$

Step 9.3.2.5

Rewrite the expression.

$x = \frac{- 17}{3} \cdot \frac{1}{- 4}$

Step 9.3.3

Multiply $\frac{- 17}{3}$ by $\frac{1}{- 4}$ .

$x = \frac{- 17}{3 \cdot - 4}$

Step 9.3.4

Multiply $3$ by $- 4$ .

$x = \frac{- 17}{- 12}$

Step 9.3.5

Dividing two negative values results in a positive value.

$x = \frac{17}{12}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{17}{12}$

Decimal Form:

$x = 1.41\overline{6}$

Mixed Number Form:

$x = 1 \frac{5}{12}$