Solve for x log of x+9- log of x=3

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

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

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

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 $b^{y} = x$ .

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

Solve for $x$

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

$1000 = \frac{x + 9}{x}$

Rewrite the equation as $\frac{x + 9}{x} = 1000$ .

$\frac{x + 9}{x} = 1000$

Solve for $x$ .

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Multiply each term by $x$ and simplify.

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Multiply each term in $\frac{x + 9}{x} = 1000$ by $x$ .

$\frac{x + 9}{x} \cdot x = 1000 \cdot x$

Cancel the common factor of $x$ .

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Cancel the common factor.

$\frac{x + 9}{x} \cdot x = 1000 \cdot x$

Rewrite the expression.

$x + 9 = 1000 \cdot x$

$x + 9 = 1000 x$

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

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

$x + 9 - 1000 x = 0$

Subtract $1000 x$ from $x$ .

$- 999 x + 9 = 0$

Subtract $9$ from both sides of the equation.

$- 999 x = - 9$

Divide each term by $- 999$ and simplify.

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Divide each term in $- 999 x = - 9$ by $- 999$ .

$\frac{- 999 x}{- 999} = \frac{- 9}{- 999}$

Cancel the common factor of $- 999$ .

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Cancel the common factor.

$\frac{- 999 x}{- 999} = \frac{- 9}{- 999}$

Divide $x$ by $1$ .

$x = \frac{- 9}{- 999}$

Cancel the common factor of $- 9$ and $- 999$ .

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

$x = \frac{- 9 \cdot 1}{- 999}$

Cancel the common factors.

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

$x = \frac{- 9 \cdot 1}{- 9 \cdot 111}$

Cancel the common factor.

$x = \frac{- 9 \cdot 1}{- 9 \cdot 111}$

Rewrite the expression.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.\overline{009}$

Solve for x log of x+9- log of x=3

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