Solve for x natural log of x-3- natural log of x-5 = natural log of 5

$\ln (x - 3) - \ln (x - 5) = \ln (5)$

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

$\ln (\frac{x - 3}{x - 5}) = \ln (5)$

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

$\frac{x - 3}{x - 5} = 5$

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.

$x - 5, 1$

Remove parentheses.

$x - 5, 1$

The LCM of one and any expression is the expression.

$x - 5$

Multiply each term in $\frac{x - 3}{x - 5} = 5$ by $x - 5$ to eliminate the fractions.

Multiply each term in $\frac{x - 3}{x - 5} = 5$ by $x - 5$ .

$\frac{x - 3}{x - 5} (x - 5) = 5 (x - 5)$

Simplify the left side.

Cancel the common factor of $x - 5$ .

Cancel the common factor.

$\frac{x - 3}{x - 5} (x - 5) = 5 (x - 5)$

Rewrite the expression.

$x - 3 = 5 (x - 5)$

Simplify the right side.

Apply the distributive property.

$x - 3 = 5 x + 5 \cdot - 5$

Multiply $5$ by $- 5$ .

$x - 3 = 5 x - 25$

Solve the equation.

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

Subtract $5 x$ from both sides of the equation.

$x - 3 - 5 x = - 25$

Subtract $5 x$ from $x$ .

$- 4 x - 3 = - 25$

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

Add $3$ to both sides of the equation.

$- 4 x = - 25 + 3$

Add $- 25$ and $3$ .

$- 4 x = - 22$

Divide each term in $- 4 x = - 22$ by $- 4$ and simplify.

Divide each term in $- 4 x = - 22$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 22}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 22}{- 4}$

Divide $x$ by $1$ .

$x = \frac{- 22}{- 4}$

Simplify the right side.

Cancel the common factor of $- 22$ and $- 4$ .

Factor $- 2$ out of $- 22$ .

$x = \frac{- 2 (11)}{- 4}$

Cancel the common factors.

Factor $- 2$ out of $- 4$ .

$x = \frac{- 2 \cdot 11}{- 2 \cdot 2}$

Cancel the common factor.

$x = \frac{- 2 \cdot 11}{- 2 \cdot 2}$

Rewrite the expression.

$x = \frac{11}{2}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{11}{2}$

Decimal Form:

$x = 5.5$

Mixed Number Form:

$x = 5 \frac{1}{2}$

Solve for x natural log of x-3- natural log of x-5 = natural log of 5

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