Solve for x natural log of x- natural log of 4x-1 = natural log of 5

$\ln (x) - \ln (4 x - 1) = \ln (5)$

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

$\ln (\frac{x}{4 x - 1}) = \ln (5)$

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

$\frac{x}{4 x - 1} = 5$

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.

$4 x - 1, 1$

Remove parentheses.

$4 x - 1, 1$

The LCM of one and any expression is the expression.

$4 x - 1$

Multiply each term in $\frac{x}{4 x - 1} = 5$ by $4 x - 1$ to eliminate the fractions.

Multiply each term in $\frac{x}{4 x - 1} = 5$ by $4 x - 1$ .

$\frac{x}{4 x - 1} (4 x - 1) = 5 (4 x - 1)$

Simplify the left side.

Cancel the common factor of $4 x - 1$ .

Cancel the common factor.

$\frac{x}{4 x - 1} (4 x - 1) = 5 (4 x - 1)$

Rewrite the expression.

$x = 5 (4 x - 1)$

Simplify the right side.

Apply the distributive property.

$x = 5 (4 x) + 5 \cdot - 1$

Multiply.

Multiply $4$ by $5$ .

$x = 20 x + 5 \cdot - 1$

Multiply $5$ by $- 1$ .

$x = 20 x - 5$

Solve the equation.

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

Subtract $20 x$ from both sides of the equation.

$x - 20 x = - 5$

Subtract $20 x$ from $x$ .

$- 19 x = - 5$

Divide each term in $- 19 x = - 5$ by $- 19$ and simplify.

Divide each term in $- 19 x = - 5$ by $- 19$ .

$\frac{- 19 x}{- 19} = \frac{- 5}{- 19}$

Simplify the left side.

Cancel the common factor of $- 19$ .

Cancel the common factor.

$\frac{- 19 x}{- 19} = \frac{- 5}{- 19}$

Divide $x$ by $1$ .

$x = \frac{- 5}{- 19}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{5}{19}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{19}$

Decimal Form:

$x = 0.26315789 \dots$

Solve for x natural log of x- natural log of 4x-1 = natural log of 5

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