Solve for x log of 5+x- log of x-2 = log of 2

$\log (5 + x) - \log (x - 2) = \log (2)$

Step 1

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

$\log (\frac{5 + x}{x - 2}) = \log (2)$

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$\frac{5 + x}{x - 2} = 2$

Step 3

Solve for $x$ .

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x - 2, 1$

Remove parentheses.

$x - 2, 1$

The LCM of one and any expression is the expression.

$x - 2$

Multiply each term in $\frac{5 + x}{x - 2} = 2$ by $x - 2$ to eliminate the fractions.

Multiply each term in $\frac{5 + x}{x - 2} = 2$ by $x - 2$ .

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

Simplify the left side.

Cancel the common factor of $x - 2$ .

Cancel the common factor.

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

Rewrite the expression.

$5 + x = 2 (x - 2)$

Simplify the right side.

Apply the distributive property.

$5 + x = 2 x + 2 \cdot - 2$

Multiply $2$ by $- 2$ .

$5 + x = 2 x - 4$

Solve the equation.

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

Subtract $2 x$ from both sides of the equation.

$5 + x - 2 x = - 4$

Subtract $2 x$ from $x$ .

$5 - x = - 4$

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

Subtract $5$ from both sides of the equation.

$- x = - 4 - 5$

Subtract $5$ from $- 4$ .

$- x = - 9$

Divide each term in $- x = - 9$ by $- 1$ and simplify.

Divide each term in $- x = - 9$ by $- 1$ .

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 9$ by $- 1$ .

$x = 9$

Solve for x log of 5+x- log of x-2 = log of 2

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