Solve for x natural log of 2- natural log of 3x+2=1

$\ln (2) - \ln (3 x + 2) = 1$

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

$\ln (\frac{2}{3 x + 2}) = 1$

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (\frac{2}{3 x + 2})} = e^{1}$

Rewrite $\ln (\frac{2}{3 x + 2}) = 1$ 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 $b^{y} = x$ .

$e^{1} = \frac{2}{3 x + 2}$

Solve for $x$ .

Rewrite the equation as $\frac{2}{3 x + 2} = e^{1}$ .

$\frac{2}{3 x + 2} = e$

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$3 x + 2, 1$

Remove parentheses.

$3 x + 2, 1$

The LCM of one and any expression is the expression.

$3 x + 2$

Multiply each term in $\frac{2}{3 x + 2} = e$ by $3 x + 2$ to eliminate the fractions.

Multiply each term in $\frac{2}{3 x + 2} = e$ by $3 x + 2$ .

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

Simplify the left side.

Cancel the common factor of $3 x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

$2 = e (3 x + 2)$

Simplify the right side.

Simplify by multiplying through.

Apply the distributive property.

$2 = e (3 x) + e \cdot 2$

Move $2$ to the left of $e$ .

$2 = e \cdot 3 x + 2 \cdot e$

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

$2 = 3 e x + 2 e$

Solve the equation.

Rewrite the equation as $3 e x + 2 e = 2$ .

$3 e x + 2 e = 2$

Subtract $2 e$ from both sides of the equation.

$3 e x = 2 - 2 e$

Divide each term in $3 e x = 2 - 2 e$ by $3 e$ and simplify.

Divide each term in $3 e x = 2 - 2 e$ by $3 e$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $e$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Simplify each term.

Cancel the common factor of $e$ .

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

$x = \frac{2}{3 e} - \frac{2}{3}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2}{3 e} - \frac{2}{3}$

Decimal Form:

$x = - 0.42141370 \dots$

Solve for x natural log of 2- natural log of 3x+2=1

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