Precalculus Examples

$\ln (x) - \ln (x - 8) = \ln (5)$

Step 1

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

$\ln (\frac{x}{x - 8}) = \ln (5)$

Step 2

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

$\frac{x}{x - 8} = 5$

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.

$x - 8, 1$

Step 3.1.2

Remove parentheses.

$x - 8, 1$

Step 3.1.3

The LCM of one and any expression is the expression.

$x - 8$

Step 3.2

Multiply each term in $\frac{x}{x - 8} = 5$ by $x - 8$ to eliminate the fractions.

Tap for more steps...

Step 3.2.1

Multiply each term in $\frac{x}{x - 8} = 5$ by $x - 8$ .

$\frac{x}{x - 8} (x - 8) = 5 (x - 8)$

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Cancel the common factor of $x - 8$ .

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor.

$\frac{x}{x - 8} (x - 8) = 5 (x - 8)$

Step 3.2.2.1.2

Rewrite the expression.

$x = 5 (x - 8)$

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Apply the distributive property.

$x = 5 x + 5 \cdot - 8$

Step 3.2.3.2

Multiply $5$ by $- 8$ .

$x = 5 x - 40$

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 $5 x$ from both sides of the equation.

$x - 5 x = - 40$

Step 3.3.1.2

Subtract $5 x$ from $x$ .

$- 4 x = - 40$

Step 3.3.2

Divide each term in $- 4 x = - 40$ by $- 4$ and simplify.

Tap for more steps...

Step 3.3.2.1

Divide each term in $- 4 x = - 40$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 40}{- 4}$

Step 3.3.2.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 40}{- 4}$

Step 3.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 40}{- 4}$

Step 3.3.2.3

Simplify the right side.

Tap for more steps...

Step 3.3.2.3.1

Divide $- 40$ by $- 4$ .

$x = 10$