Solve for x log base y of x+2- log base y of x-1 = log base y of 4

$\log_{y} (x + 2) - \log_{y} (x - 1) = \log_{y} (4)$

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

$\log_{y} (\frac{x + 2}{x - 1}) = \log_{y} (4)$

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

$\frac{x + 2}{x - 1} = 4$

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.

$x - 1, 1$

Remove parentheses.

$x - 1, 1$

The LCM of one and any expression is the expression.

$x - 1$

Multiply each term in $\frac{x + 2}{x - 1} = 4$ by $x - 1$ to eliminate the fractions.

Multiply each term in $\frac{x + 2}{x - 1} = 4$ by $x - 1$ .

$\frac{x + 2}{x - 1} (x - 1) = 4 (x - 1)$

Simplify the left side.

Cancel the common factor of $x - 1$ .

Cancel the common factor.

$\frac{x + 2}{x - 1} (x - 1) = 4 (x - 1)$

Rewrite the expression.

$x + 2 = 4 (x - 1)$

Simplify the right side.

Apply the distributive property.

$x + 2 = 4 x + 4 \cdot - 1$

Multiply $4$ by $- 1$ .

$x + 2 = 4 x - 4$

Solve the equation.

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

Subtract $4 x$ from both sides of the equation.

$x + 2 - 4 x = - 4$

Subtract $4 x$ from $x$ .

$- 3 x + 2 = - 4$

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

Subtract $2$ from both sides of the equation.

$- 3 x = - 4 - 2$

Subtract $2$ from $- 4$ .

$- 3 x = - 6$

Divide each term in $- 3 x = - 6$ by $- 3$ and simplify.

Divide each term in $- 3 x = - 6$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 6}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 6}{- 3}$

Divide $x$ by $1$ .

$x = \frac{- 6}{- 3}$

Simplify the right side.

Divide $- 6$ by $- 3$ .

$x = 2$

Solve for x log base y of x+2- log base y of x-1 = log base y of 4

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