Evaluate natural log of x-8+ natural log of x+1 = natural log of x-24

$\ln (x - 8) + \ln (x + 1) = \ln (x - 24)$

Step 1

Simplify the left side.

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

$\ln ((x - 8) (x + 1)) = \ln (x - 24)$

Expand $(x - 8) (x + 1)$ using the FOIL Method.

Apply the distributive property.

$\ln (x (x + 1) - 8 (x + 1)) = \ln (x - 24)$

Apply the distributive property.

$\ln (x \cdot x + x \cdot 1 - 8 (x + 1)) = \ln (x - 24)$

Apply the distributive property.

$\ln (x \cdot x + x \cdot 1 - 8 x - 8 \cdot 1) = \ln (x - 24)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\ln (x^{2} + x \cdot 1 - 8 x - 8 \cdot 1) = \ln (x - 24)$

Multiply $x$ by $1$ .

$\ln (x^{2} + x - 8 x - 8 \cdot 1) = \ln (x - 24)$

Multiply $- 8$ by $1$ .

$\ln (x^{2} + x - 8 x - 8) = \ln (x - 24)$

Subtract $8 x$ from $x$ .

$\ln (x^{2} - 7 x - 8) = \ln (x - 24)$

Step 2

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

$x^{2} - 7 x - 8 = x - 24$

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.

$x^{2} - 7 x - 8 - x = - 24$

Subtract $x$ from $- 7 x$ .

$x^{2} - 8 x - 8 = - 24$

Add $24$ to both sides of the equation.

$x^{2} - 8 x - 8 + 24 = 0$

Add $- 8$ and $24$ .

$x^{2} - 8 x + 16 = 0$

Factor using the perfect square rule.

Rewrite $16$ as $4^{2}$ .

$x^{2} - 8 x + 4^{2} = 0$

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

$8 x = 2 \cdot x \cdot 4$

Rewrite the polynomial.

$x^{2} - 2 \cdot x \cdot 4 + 4^{2} = 0$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 4$ .

${(x - 4)}^{2} = 0$

Set the $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

Step 4

Exclude the solutions that do not make $\ln (x - 8) + \ln (x + 1) = \ln (x - 24)$ true.

$No solution$

Evaluate natural log of x-8+ natural log of x+1 = natural log of x-24

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