Algebra Examples

$\log (x + 9) - \log (x) = 3$

Step 1

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

$\log (\frac{x + 9}{x}) = 3$

Step 2

Rewrite $\log (\frac{x + 9}{x}) = 3$ 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 $b^{y} = x$ .

$10^{3} = \frac{x + 9}{x}$

Step 3

Cross multiply to remove the fraction.

$x + 9 = 10^{3} (x)$

Step 4

Raise $10$ to the power of $3$ .

$x + 9 = 1000 x$

Step 5

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

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Step 5.1

Subtract $1000 x$ from both sides of the equation.

$x + 9 - 1000 x = 0$

Step 5.2

Subtract $1000 x$ from $x$ .

$- 999 x + 9 = 0$

Step 6

Factor $9$ out of $- 999 x + 9$ .

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Step 6.1

Factor $9$ out of $- 999 x$ .

$9 (- 111 x) + 9 = 0$

Step 6.2

Factor $9$ out of $9$ .

$9 (- 111 x) + 9 (1) = 0$

Step 6.3

Factor $9$ out of $9 (- 111 x) + 9 (1)$ .

$9 (- 111 x + 1) = 0$

Step 7

Divide each term in $9 (- 111 x + 1) = 0$ by $9$ and simplify.

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Step 7.1

Divide each term in $9 (- 111 x + 1) = 0$ by $9$ .

$\frac{9 (- 111 x + 1)}{9} = \frac{0}{9}$

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.

$\frac{9 (- 111 x + 1)}{9} = \frac{0}{9}$

Step 7.2.1.2

Divide $- 111 x + 1$ by $1$ .

$- 111 x + 1 = \frac{0}{9}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $0$ by $9$ .

$- 111 x + 1 = 0$

Step 8

Subtract $1$ from both sides of the equation.

$- 111 x = - 1$

Step 9

Divide each term in $- 111 x = - 1$ by $- 111$ and simplify.

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Step 9.1

Divide each term in $- 111 x = - 1$ by $- 111$ .

$\frac{- 111 x}{- 111} = \frac{- 1}{- 111}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $- 111$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{- 111 x}{- 111} = \frac{- 1}{- 111}$

Step 9.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 111}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Dividing two negative values results in a positive value.

$x = \frac{1}{111}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{111}$

Decimal Form:

$x = 0.\overline{009}$