Precalculus Examples

$\log_{8} (x + 5) - \log_{8} (x - 2) = 1$

Step 1

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

$\log_{8} (\frac{x + 5}{x - 2}) = 1$

Step 2

Rewrite $\log_{8} (\frac{x + 5}{x - 2}) = 1$ 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 $b^{y} = x$ .

$8^{1} = \frac{x + 5}{x - 2}$

Step 3

Cross multiply to remove the fraction.

$x + 5 = 8 (x - 2)$

Step 4

Simplify $8 (x - 2)$ .

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Step 4.1

Apply the distributive property.

$x + 5 = 8 x + 8 \cdot - 2$

Step 4.2

Multiply $8$ by $- 2$ .

$x + 5 = 8 x - 16$

Step 5

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

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Step 5.1

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

$x + 5 - 8 x = - 16$

Step 5.2

Subtract $8 x$ from $x$ .

$- 7 x + 5 = - 16$

Step 6

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

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Step 6.1

Subtract $5$ from both sides of the equation.

$- 7 x = - 16 - 5$

Step 6.2

Subtract $5$ from $- 16$ .

$- 7 x = - 21$

Step 7

Divide each term in $- 7 x = - 21$ by $- 7$ and simplify.

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Step 7.1

Divide each term in $- 7 x = - 21$ by $- 7$ .

$\frac{- 7 x}{- 7} = \frac{- 21}{- 7}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 7$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{- 7 x}{- 7} = \frac{- 21}{- 7}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 21}{- 7}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $- 21$ by $- 7$ .

$x = 3$