Solve for x log base 3 of x-3- log base 3 of 2x+3=2

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

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{3} (\frac{x - 3}{2 x + 3}) = 2$

Rewrite $\log_{3} (\frac{x - 3}{2 x + 3}) = 2$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ $\neq$ $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$3^{2} = \frac{x - 3}{2 x + 3}$

Cross multiply to remove the fraction.

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

Simplify $3^{2} (2 x + 3)$ .

Raise $3$ to the power of $2$ .

$x - 3 = 9 (2 x + 3)$

Apply the distributive property.

$x - 3 = 9 (2 x) + 9 \cdot 3$

Multiply.

Multiply $2$ by $9$ .

$x - 3 = 18 x + 9 \cdot 3$

Multiply $9$ by $3$ .

$x - 3 = 18 x + 27$

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

Subtract $18 x$ from both sides of the equation.

$x - 3 - 18 x = 27$

Subtract $18 x$ from $x$ .

$- 17 x - 3 = 27$

Move all terms not containing $x$ to the right side of the equation.

Add $3$ to both sides of the equation.

$- 17 x = 27 + 3$

Add $27$ and $3$ .

$- 17 x = 30$

Divide each term in $- 17 x = 30$ by $- 17$ and simplify.

Divide each term in $- 17 x = 30$ by $- 17$ .

$\frac{- 17 x}{- 17} = \frac{30}{- 17}$

Simplify the left side.

Cancel the common factor of $- 17$ .

Cancel the common factor.

$\frac{- 17 x}{- 17} = \frac{30}{- 17}$

Divide $x$ by $1$ .

$x = \frac{30}{- 17}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{30}{17}$

Exclude the solutions that do not make $\log_{3} (x - 3) - \log_{3} (2 x + 3) = 2$ true.

$No solution$

Solve for x log base 3 of x-3- log base 3 of 2x+3=2

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