Precalculus Examples

$\ln (x - 4) - \ln (x + 1) = \ln (6)$

Step 1

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

$\ln (\frac{x - 4}{x + 1}) = \ln (6)$

Step 2

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

$\frac{x - 4}{x + 1} = 6$

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Find the LCD of the terms in the equation.

Tap for more steps...

Step 3.1.1

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

$x + 1, 1$

Step 3.1.2

Remove parentheses.

$x + 1, 1$

Step 3.1.3

The LCM of one and any expression is the expression.

$x + 1$

Step 3.2

Multiply each term in $\frac{x - 4}{x + 1} = 6$ by $x + 1$ to eliminate the fractions.

Tap for more steps...

Step 3.2.1

Multiply each term in $\frac{x - 4}{x + 1} = 6$ by $x + 1$ .

$\frac{x - 4}{x + 1} (x + 1) = 6 (x + 1)$

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Cancel the common factor of $x + 1$ .

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor.

$\frac{x - 4}{x + 1} (x + 1) = 6 (x + 1)$

Step 3.2.2.1.2

Rewrite the expression.

$x - 4 = 6 (x + 1)$

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Apply the distributive property.

$x - 4 = 6 x + 6 \cdot 1$

Step 3.2.3.2

Multiply $6$ by $1$ .

$x - 4 = 6 x + 6$

Step 3.3

Solve the equation.

Tap for more steps...

Step 3.3.1

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

Tap for more steps...

Step 3.3.1.1

Subtract $6 x$ from both sides of the equation.

$x - 4 - 6 x = 6$

Step 3.3.1.2

Subtract $6 x$ from $x$ .

$- 5 x - 4 = 6$

Step 3.3.2

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

Tap for more steps...

Step 3.3.2.1

Add $4$ to both sides of the equation.

$- 5 x = 6 + 4$

Step 3.3.2.2

Add $6$ and $4$ .

$- 5 x = 10$

Step 3.3.3

Divide each term in $- 5 x = 10$ by $- 5$ and simplify.

Tap for more steps...

Step 3.3.3.1

Divide each term in $- 5 x = 10$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{10}{- 5}$

Step 3.3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.3.2.1

Cancel the common factor of $- 5$ .

Tap for more steps...

Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{10}{- 5}$

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{10}{- 5}$

Step 3.3.3.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.3.1

Divide $10$ by $- 5$ .

$x = - 2$

Step 4

Exclude the solutions that do not make $\ln (x - 4) - \ln (x + 1) = \ln (6)$ true.

$No solution$