Solve for x log base 5 of x+16- log base 5 of x-8=2

$\log_{5} (x + 16) - \log_{5} (x - 8) = 2$

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

$\log_{5} (\frac{x + 16}{x - 8}) = 2$

Rewrite $\log_{5} (\frac{x + 16}{x - 8}) = 2$ 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 $b^{y} = x$ .

$5^{2} = \frac{x + 16}{x - 8}$

Cross multiply to remove the fraction.

$x + 16 = 5^{2} (x - 8)$

Simplify $5^{2} (x - 8)$ .

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

$x + 16 = 25 (x - 8)$

Apply the distributive property.

$x + 16 = 25 x + 25 \cdot - 8$

Multiply $25$ by $- 8$ .

$x + 16 = 25 x - 200$

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

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

$x + 16 - 25 x = - 200$

Subtract $25 x$ from $x$ .

$- 24 x + 16 = - 200$

Factor $8$ out of $- 24 x + 16$ .

Factor $8$ out of $- 24 x$ .

$8 (- 3 x) + 16 = - 200$

Factor $8$ out of $16$ .

$8 (- 3 x) + 8 (2) = - 200$

Factor $8$ out of $8 (- 3 x) + 8 (2)$ .

$8 (- 3 x + 2) = - 200$

Divide each term in $8 (- 3 x + 2) = - 200$ by $8$ and simplify.

Divide each term in $8 (- 3 x + 2) = - 200$ by $8$ .

$\frac{8 (- 3 x + 2)}{8} = \frac{- 200}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 (- 3 x + 2)}{8} = \frac{- 200}{8}$

Divide $- 3 x + 2$ by $1$ .

$- 3 x + 2 = \frac{- 200}{8}$

Simplify the right side.

Divide $- 200$ by $8$ .

$- 3 x + 2 = - 25$

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

Subtract $2$ from both sides of the equation.

$- 3 x = - 25 - 2$

Subtract $2$ from $- 25$ .

$- 3 x = - 27$

Divide each term in $- 3 x = - 27$ by $- 3$ and simplify.

Divide each term in $- 3 x = - 27$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 27}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 27}{- 3}$

Divide $x$ by $1$ .

$x = \frac{- 27}{- 3}$

Simplify the right side.

Divide $- 27$ by $- 3$ .

$x = 9$

Solve for x log base 5 of x+16- log base 5 of x-8=2

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