Solve for x natural log of x-4+ natural log of x+3 = natural log of x-13

$\ln (x - 4) + \ln (x + 3) = \ln (x - 13)$

Simplify the left side.

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

$\ln ((x - 4) (x + 3)) = \ln (x - 13)$

Expand $(x - 4) (x + 3)$ using the FOIL Method.

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Apply the distributive property.

$\ln (x (x + 3) - 4 (x + 3)) = \ln (x - 13)$

Apply the distributive property.

$\ln (x \cdot x + x \cdot 3 - 4 (x + 3)) = \ln (x - 13)$

Apply the distributive property.

$\ln (x \cdot x + x \cdot 3 - 4 x - 4 \cdot 3) = \ln (x - 13)$

Simplify and combine like terms.

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Simplify each term.

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Multiply $x$ by $x$ .

$\ln (x^{2} + x \cdot 3 - 4 x - 4 \cdot 3) = \ln (x - 13)$

Move $3$ to the left of $x$ .

$\ln (x^{2} + 3 \cdot x - 4 x - 4 \cdot 3) = \ln (x - 13)$

Multiply $- 4$ by $3$ .

$\ln (x^{2} + 3 x - 4 x - 12) = \ln (x - 13)$

Subtract $4 x$ from $3 x$ .

$\ln (x^{2} - x - 12) = \ln (x - 13)$

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

$x^{2} - x - 12 = x - 13$

Solve for $x$ .

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

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Subtract $x$ from both sides of the equation.

$x^{2} - x - 12 - x = - 13$

Subtract $x$ from $- x$ .

$x^{2} - 2 x - 12 = - 13$

Move $13$ to the left side of the equation by adding it to both sides.

$x^{2} - 2 x - 12 + 13 = 0$

Add $- 12$ and $13$ .

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

Factor using the perfect square rule.

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

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

Check the middle term by multiplying $2 a b$ and compare this result with the middle term in the original expression.

$2 a b = 2 \cdot x \cdot - 1$

Simplify.

$2 a b = - 2 x$

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 $x - 1$ equal to $0$ and solve for $x$ .

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Set the factor equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

The solution is the result of $x - 1 = 0$ .

$x = 1$

Exclude the solutions that do not make $\ln (x - 4) + \ln (x + 3) = \ln (x - 13)$ true.

No solution

Solve for x natural log of x-4+ natural log of x+3 = natural log of x-13

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