Solve for x natural log of x- natural log of 3x-3 = natural log of 4

$\ln (x) - \ln (3 x - 3) = \ln (4)$

Step 1

Simplify the left side.

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

$\ln (\frac{x}{3 x - 3}) = \ln (4)$

Factor $3$ out of $3 x - 3$ .

Factor $3$ out of $3 x$ .

$\ln (\frac{x}{3 (x) - 3}) = \ln (4)$

Factor $3$ out of $- 3$ .

$\ln (\frac{x}{3 (x) + 3 (- 1)}) = \ln (4)$

Factor $3$ out of $3 (x) + 3 (- 1)$ .

$\ln (\frac{x}{3 (x - 1)}) = \ln (4)$

Step 2

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

$\frac{x}{3 (x - 1)} = 4$

Step 3

Solve for $x$ .

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.

$3 (x - 1), 1$

The LCM of one and any expression is the expression.

$3 (x - 1)$

Multiply each term in $\frac{x}{3 (x - 1)} = 4$ by $3 (x - 1)$ to eliminate the fractions.

Multiply each term in $\frac{x}{3 (x - 1)} = 4$ by $3 (x - 1)$ .

$\frac{x}{3 (x - 1)} (3 (x - 1)) = 4 (3 (x - 1))$

Simplify the left side.

Rewrite using the commutative property of multiplication.

$3 \frac{x}{3 (x - 1)} (x - 1) = 4 (3 (x - 1))$

Cancel the common factor of $3$ .

Cancel the common factor.

$3 \frac{x}{3 (x - 1)} (x - 1) = 4 (3 (x - 1))$

Rewrite the expression.

$\frac{x}{x - 1} (x - 1) = 4 (3 (x - 1))$

Cancel the common factor of $x - 1$ .

Cancel the common factor.

$\frac{x}{x - 1} (x - 1) = 4 (3 (x - 1))$

Rewrite the expression.

$x = 4 (3 (x - 1))$

Simplify the right side.

Apply the distributive property.

$x = 4 (3 x + 3 \cdot - 1)$

Multiply $3$ by $- 1$ .

$x = 4 (3 x - 3)$

Apply the distributive property.

$x = 4 (3 x) + 4 \cdot - 3$

Multiply.

Multiply $3$ by $4$ .

$x = 12 x + 4 \cdot - 3$

Multiply $4$ by $- 3$ .

$x = 12 x - 12$

Solve the equation.

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

Subtract $12 x$ from both sides of the equation.

$x - 12 x = - 12$

Subtract $12 x$ from $x$ .

$- 11 x = - 12$

Divide each term in $- 11 x = - 12$ by $- 11$ and simplify.

Divide each term in $- 11 x = - 12$ by $- 11$ .

$\frac{- 11 x}{- 11} = \frac{- 12}{- 11}$

Simplify the left side.

Cancel the common factor of $- 11$ .

Cancel the common factor.

$\frac{- 11 x}{- 11} = \frac{- 12}{- 11}$

Divide $x$ by $1$ .

$x = \frac{- 12}{- 11}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{12}{11}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{12}{11}$

Decimal Form:

$x = 1.\overline{09}$

Mixed Number Form:

$x = 1 \frac{1}{11}$

Solve for x natural log of x- natural log of 3x-3 = natural log of 4

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