Solve for x 2 log of x+1/2 log of 9 = log of 3x+4+ log of x-2

$2 \log (x) + \frac{1}{2} \log (9) = \log (3 x + 4) + \log (x - 2)$

Simplify the left side.

Combine $\frac{1}{2}$ and $\log (9)$ .

$2 \log (x) + \frac{\log (9)}{2} = \log (3 x + 4) + \log (x - 2)$

Simplify the left side.

Simplify $2 \log (x) + \frac{\log (9)}{2}$ .

Simplify each term.

Simplify $2 \log (x)$ by moving $2$ inside the logarithm.

$\log (x^{2}) + \frac{\log (9)}{2} = \log (3 x + 4) + \log (x - 2)$

Rewrite $\log (9)$ as $\log (3^{2})$ .

$\log (x^{2}) + \frac{\log (3^{2})}{2} = \log (3 x + 4) + \log (x - 2)$

Expand $\log (3^{2})$ by moving $2$ outside the logarithm.

$\log (x^{2}) + \frac{2 \log (3)}{2} = \log (3 x + 4) + \log (x - 2)$

Cancel the common factor of $2$ .

Cancel the common factor.

$\log (x^{2}) + \frac{2 \log (3)}{2} = \log (3 x + 4) + \log (x - 2)$

Divide $\log (3)$ by $1$ .

$\log (x^{2}) + \log (3) = \log (3 x + 4) + \log (x - 2)$

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

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

Move $3$ to the left of $x^{2}$ .

$\log (3 x^{2}) = \log (3 x + 4) + \log (x - 2)$

Simplify the right side.

Simplify $\log (3 x + 4) + \log (x - 2)$ .

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

$\log (3 x^{2}) = \log ((3 x + 4) (x - 2))$

Expand $(3 x + 4) (x - 2)$ using the FOIL Method.

Apply the distributive property.

$\log (3 x^{2}) = \log (3 x (x - 2) + 4 (x - 2))$

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $3$ .

$\log (3 x^{2}) = \log (3 x^{2} - 6 x + 4 x + 4 \cdot - 2)$

Multiply $4$ by $- 2$ .

$\log (3 x^{2}) = \log (3 x^{2} - 6 x + 4 x - 8)$

Add $- 6 x$ and $4 x$ .

$\log (3 x^{2}) = \log (3 x^{2} - 2 x - 8)$

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

$3 x^{2} = 3 x^{2} - 2 x - 8$

Solve for $x$ .

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$3 x^{2} - 2 x - 8 = 3 x^{2}$

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

Subtract $3 x^{2}$ from both sides of the equation.

$3 x^{2} - 2 x - 8 - 3 x^{2} = 0$

Combine the opposite terms in $3 x^{2} - 2 x - 8 - 3 x^{2}$ .

Subtract $3 x^{2}$ from $3 x^{2}$ .

$- 2 x - 8 + 0 = 0$

Add $- 2 x - 8$ and $0$ .

$- 2 x - 8 = 0$

Add $8$ to both sides of the equation.

$- 2 x = 8$

Divide each term in $- 2 x = 8$ by $- 2$ and simplify.

Divide each term in $- 2 x = 8$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{8}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{8}{- 2}$

Divide $x$ by $1$ .

$x = \frac{8}{- 2}$

Simplify the right side.

Divide $8$ by $- 2$ .

$x = - 4$

Exclude the solutions that do not make $2 \log (x) + \frac{1}{2} \cdot \log (9) = \log (3 x + 4) + \log (x - 2)$ true.

$No solution$

Solve for x 2 log of x+1/2 log of 9 = log of 3x+4+ log of x-2

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