Precalculus Examples

$\log_{3} (x - 2) = \log_{3} (x + 6) + 2$

Step 1

Move all the terms containing a logarithm to the left side of the equation.

$\log_{3} (x - 2) - \log_{3} (x + 6) = 2$

Step 2

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

$\log_{3} (\frac{x - 2}{x + 6}) = 2$

Step 3

Rewrite $\log_{3} (\frac{x - 2}{x + 6}) = 2$ 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 $b^{y} = x$ .

$3^{2} = \frac{x - 2}{x + 6}$

Step 4

Cross multiply to remove the fraction.

$x - 2 = 3^{2} (x + 6)$

Step 5

Simplify $3^{2} (x + 6)$ .

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Step 5.1

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

$x - 2 = 9 (x + 6)$

Step 5.2

Apply the distributive property.

$x - 2 = 9 x + 9 \cdot 6$

Step 5.3

Multiply $9$ by $6$ .

$x - 2 = 9 x + 54$

Step 6

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

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Step 6.1

Subtract $9 x$ from both sides of the equation.

$x - 2 - 9 x = 54$

Step 6.2

Subtract $9 x$ from $x$ .

$- 8 x - 2 = 54$

Step 7

Factor $2$ out of $- 8 x - 2$ .

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Step 7.1

Factor $2$ out of $- 8 x$ .

$2 (- 4 x) - 2 = 54$

Step 7.2

Factor $2$ out of $- 2$ .

$2 (- 4 x) + 2 (- 1) = 54$

Step 7.3

Factor $2$ out of $2 (- 4 x) + 2 (- 1)$ .

$2 (- 4 x - 1) = 54$

Step 8

Divide each term in $2 (- 4 x - 1) = 54$ by $2$ and simplify.

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Step 8.1

Divide each term in $2 (- 4 x - 1) = 54$ by $2$ .

$\frac{2 (- 4 x - 1)}{2} = \frac{54}{2}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $2$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{2 (- 4 x - 1)}{2} = \frac{54}{2}$

Step 8.2.1.2

Divide $- 4 x - 1$ by $1$ .

$- 4 x - 1 = \frac{54}{2}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Divide $54$ by $2$ .

$- 4 x - 1 = 27$

Step 9

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

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Step 9.1

Add $1$ to both sides of the equation.

$- 4 x = 27 + 1$

Step 9.2

Add $27$ and $1$ .

$- 4 x = 28$

Step 10

Divide each term in $- 4 x = 28$ by $- 4$ and simplify.

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Step 10.1

Divide each term in $- 4 x = 28$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{28}{- 4}$

Step 10.2

Simplify the left side.

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Step 10.2.1

Cancel the common factor of $- 4$ .

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Step 10.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{28}{- 4}$

Step 10.2.1.2

Divide $x$ by $1$ .

$x = \frac{28}{- 4}$

Step 10.3

Simplify the right side.

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Step 10.3.1

Divide $28$ by $- 4$ .

$x = - 7$

Step 11

Exclude the solutions that do not make $\log_{3} (x - 2) = \log_{3} (x + 6) + 2$ true.

$No solution$