Solve for x natural log of x+31- natural log of 4-3x=5 natural log of 2

$\ln (x + 31) - \ln (4 - 3 x) = 5 \ln (2)$

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

$\ln (\frac{x + 31}{4 - 3 x}) = 5 \ln (2)$

Simplify $5 \ln (2)$ by moving $5$ inside the logarithm.

$\ln (\frac{x + 31}{4 - 3 x}) = \ln (2^{5})$

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

$\frac{x + 31}{4 - 3 x} = 2^{5}$

Solve for $x$ .

Raise $2$ to the power of $5$ .

$\frac{x + 31}{4 - 3 x} = 32$

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.

$4 - 3 x, 1$

Remove parentheses.

$4 - 3 x, 1$

The LCM of one and any expression is the expression.

$4 - 3 x$

Multiply each term in $\frac{x + 31}{4 - 3 x} = 32$ by $4 - 3 x$ to eliminate the fractions.

Multiply each term in $\frac{x + 31}{4 - 3 x} = 32$ by $4 - 3 x$ .

$\frac{x + 31}{4 - 3 x} (4 - 3 x) = 32 (4 - 3 x)$

Simplify the left side.

Cancel the common factor of $4 - 3 x$ .

Cancel the common factor.

$\frac{x + 31}{4 - 3 x} (4 - 3 x) = 32 (4 - 3 x)$

Rewrite the expression.

$x + 31 = 32 (4 - 3 x)$

Simplify the right side.

Apply the distributive property.

$x + 31 = 32 \cdot 4 + 32 (- 3 x)$

Multiply.

Multiply $32$ by $4$ .

$x + 31 = 128 + 32 (- 3 x)$

Multiply $- 3$ by $32$ .

$x + 31 = 128 - 96 x$

Solve the equation.

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

Add $96 x$ to both sides of the equation.

$x + 31 + 96 x = 128$

Add $x$ and $96 x$ .

$97 x + 31 = 128$

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

Subtract $31$ from both sides of the equation.

$97 x = 128 - 31$

Subtract $31$ from $128$ .

$97 x = 97$

Divide each term in $97 x = 97$ by $97$ and simplify.

Divide each term in $97 x = 97$ by $97$ .

$\frac{97 x}{97} = \frac{97}{97}$

Simplify the left side.

Cancel the common factor of $97$ .

Cancel the common factor.

$\frac{97 x}{97} = \frac{97}{97}$

Divide $x$ by $1$ .

$x = \frac{97}{97}$

Simplify the right side.

Divide $97$ by $97$ .

$x = 1$

Solve for x natural log of x+31- natural log of 4-3x=5 natural log of 2

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