Solve for x log base 5 of x+17- log base 5 of x-7=2

$\log_{5} (x + 17) - \log_{5} (x - 7) = 2$

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

$\log_{5} (\frac{x + 17}{x - 7}) = 2$

Rewrite $\log_{5} (\frac{x + 17}{x - 7}) = 2$ 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 $b^{y} = x$ .

$5^{2} = \frac{x + 17}{x - 7}$

Solve for $x$

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Raise $5$ to the power of $2$ .

$25 = \frac{x + 17}{x - 7}$

Rewrite the equation as $\frac{x + 17}{x - 7} = 25$ .

$\frac{x + 17}{x - 7} = 25$

Solve for $x$ .

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Multiply each term by $x - 7$ and simplify.

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Multiply each term in $\frac{x + 17}{x - 7} = 25$ by $x - 7$ .

$\frac{x + 17}{x - 7} \cdot (x - 7) = 25 \cdot (x - 7)$

Cancel the common factor of $x - 7$ .

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Cancel the common factor.

$\frac{x + 17}{x - 7} \cdot (x - 7) = 25 \cdot (x - 7)$

Rewrite the expression.

$x + 17 = 25 \cdot (x - 7)$

Simplify $25 \cdot (x - 7)$ .

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Apply the distributive property.

$x + 17 = 25 x + 25 \cdot - 7$

Multiply $25$ by $- 7$ .

$x + 17 = 25 x - 175$

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

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Subtract $25 x$ from both sides of the equation.

$x + 17 - 25 x = - 175$

Subtract $25 x$ from $x$ .

$- 24 x + 17 = - 175$

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

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Subtract $17$ from both sides of the equation.

$- 24 x = - 175 - 17$

Subtract $17$ from $- 175$ .

$- 24 x = - 192$

Divide each term by $- 24$ and simplify.

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Divide each term in $- 24 x = - 192$ by $- 24$ .

$\frac{- 24 x}{- 24} = \frac{- 192}{- 24}$

Cancel the common factor of $- 24$ .

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Cancel the common factor.

$\frac{- 24 x}{- 24} = \frac{- 192}{- 24}$

Divide $x$ by $1$ .

$x = \frac{- 192}{- 24}$

Divide $- 192$ by $- 24$ .

$x = 8$

Solve for x log base 5 of x+17- log base 5 of x-7=2

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