Calculus Examples

$\log (x + 9) - \log (x - 9) = 1$

Step 1

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

$\log (\frac{x + 9}{x - 9}) = 1$

Step 2

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

$10^{1} = \frac{x + 9}{x - 9}$

Step 3

Cross multiply to remove the fraction.

$x + 9 = 10 (x - 9)$

Step 4

Simplify $10 (x - 9)$ .

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Step 4.1

Apply the distributive property.

$x + 9 = 10 x + 10 \cdot - 9$

Step 4.2

Multiply $10$ by $- 9$ .

$x + 9 = 10 x - 90$

Step 5

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

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Step 5.1

Subtract $10 x$ from both sides of the equation.

$x + 9 - 10 x = - 90$

Step 5.2

Subtract $10 x$ from $x$ .

$- 9 x + 9 = - 90$

Step 6

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

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Step 6.1

Factor $9$ out of $- 9 x$ .

$9 (- x) + 9 = - 90$

Step 6.2

Factor $9$ out of $9$ .

$9 (- x) + 9 (1) = - 90$

Step 6.3

Factor $9$ out of $9 (- x) + 9 (1)$ .

$9 (- x + 1) = - 90$

Step 7

Divide each term in $9 (- x + 1) = - 90$ by $9$ and simplify.

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Step 7.1

Divide each term in $9 (- x + 1) = - 90$ by $9$ .

$\frac{9 (- x + 1)}{9} = \frac{- 90}{9}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $9$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{9 (- x + 1)}{9} = \frac{- 90}{9}$

Step 7.2.1.2

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

$- x + 1 = \frac{- 90}{9}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $- 90$ by $9$ .

$- x + 1 = - 10$

Step 8

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

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Step 8.1

Subtract $1$ from both sides of the equation.

$- x = - 10 - 1$

Step 8.2

Subtract $1$ from $- 10$ .

$- x = - 11$

Step 9

Divide each term in $- x = - 11$ by $- 1$ and simplify.

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Step 9.1

Divide each term in $- x = - 11$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 11}{- 1}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 11}{- 1}$

Step 9.2.2

Divide $x$ by $1$ .

$x = \frac{- 11}{- 1}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Divide $- 11$ by $- 1$ .

$x = 11$