Solve for x natural log of x-2+ natural log of x+1 = natural log of x-3

$\ln (x - 2) + \ln (x + 1) = \ln (x - 3)$

Step 1

Simplify the left side.

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln ((x - 2) (x + 1)) = \ln (x - 3)$

Expand $(x - 2) (x + 1)$ using the FOIL Method.

Apply the distributive property.

$\ln (x (x + 1) - 2 (x + 1)) = \ln (x - 3)$

Apply the distributive property.

$\ln (x \cdot x + x \cdot 1 - 2 (x + 1)) = \ln (x - 3)$

Apply the distributive property.

$\ln (x \cdot x + x \cdot 1 - 2 x - 2 \cdot 1) = \ln (x - 3)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\ln (x^{2} + x \cdot 1 - 2 x - 2 \cdot 1) = \ln (x - 3)$

Multiply $x$ by $1$ .

$\ln (x^{2} + x - 2 x - 2 \cdot 1) = \ln (x - 3)$

Multiply $- 2$ by $1$ .

$\ln (x^{2} + x - 2 x - 2) = \ln (x - 3)$

Subtract $2 x$ from $x$ .

$\ln (x^{2} - x - 2) = \ln (x - 3)$

Step 2

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

$x^{2} - x - 2 = x - 3$

Step 3

Solve for $x$ .

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

Subtract $x$ from both sides of the equation.

$x^{2} - x - 2 - x = - 3$

Subtract $x$ from $- x$ .

$x^{2} - 2 x - 2 = - 3$

Add $3$ to both sides of the equation.

$x^{2} - 2 x - 2 + 3 = 0$

Add $- 2$ and $3$ .

$x^{2} - 2 x + 1 = 0$

Factor using the perfect square rule.

Rewrite $1$ as $1^{2}$ .

$x^{2} - 2 x + 1^{2} = 0$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Rewrite the polynomial.

$x^{2} - 2 \cdot x \cdot 1 + 1^{2} = 0$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

${(x - 1)}^{2} = 0$

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Step 4

Exclude the solutions that do not make $\ln (x - 2) + \ln (x + 1) = \ln (x - 3)$ true.

$No solution$

Solve for x natural log of x-2+ natural log of x+1 = natural log of x-3

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