Solve for x natural log of x-2+ natural log of x-6 = natural log of 5x

$\ln (x - 2) + \ln (x - 6) = \ln (5 x)$

Simplify the left side.

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

$\ln ((x - 2) (x - 6)) = \ln (5 x)$

Expand $(x - 2) (x - 6)$ using the FOIL Method.

Apply the distributive property.

$\ln (x (x - 6) - 2 (x - 6)) = \ln (5 x)$

Apply the distributive property.

$\ln (x \cdot x + x \cdot - 6 - 2 (x - 6)) = \ln (5 x)$

Apply the distributive property.

$\ln (x \cdot x + x \cdot - 6 - 2 x - 2 \cdot - 6) = \ln (5 x)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\ln (x^{2} + x \cdot - 6 - 2 x - 2 \cdot - 6) = \ln (5 x)$

Move $- 6$ to the left of $x$ .

$\ln (x^{2} - 6 \cdot x - 2 x - 2 \cdot - 6) = \ln (5 x)$

Multiply $- 2$ by $- 6$ .

$\ln (x^{2} - 6 x - 2 x + 12) = \ln (5 x)$

Subtract $2 x$ from $- 6 x$ .

$\ln (x^{2} - 8 x + 12) = \ln (5 x)$

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

$x^{2} - 8 x + 12 = 5 x$

Solve for $x$ .

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

Subtract $5 x$ from both sides of the equation.

$x^{2} - 8 x + 12 - 5 x = 0$

Subtract $5 x$ from $- 8 x$ .

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

Factor $x^{2} - 13 x + 12$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $12$ and whose sum is $- 13$ .

$- 12, - 1$

Write the factored form using these integers.

$(x - 12) (x - 1) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 12 = 0$

$x - 1 = 0$

Set $x - 12$ equal to $0$ and solve for $x$ .

Set $x - 12$ equal to $0$ .

$x - 12 = 0$

Add $12$ to both sides of the equation.

$x = 12$

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

The final solution is all the values that make $(x - 12) (x - 1) = 0$ true.

$x = 12, 1$

Exclude the solutions that do not make $\ln (x - 2) + \ln (x - 6) = \ln (5 x)$ true.

$x = 12$

Solve for x natural log of x-2+ natural log of x-6 = natural log of 5x

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