Precalculus Examples

$\log_{6} (x + 31) - \log_{6} (x - 4) = 2$

Step 1

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

$\log_{6} (\frac{x + 31}{x - 4}) = 2$

Step 2

Rewrite $\log_{6} (\frac{x + 31}{x - 4}) = 2$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ $\neq$ $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$6^{2} = \frac{x + 31}{x - 4}$

Step 3

Cross multiply to remove the fraction.

$x + 31 = 6^{2} (x - 4)$

Step 4

Simplify $6^{2} (x - 4)$ .

Tap for more steps...

Step 4.1

Raise $6$ to the power of $2$ .

$x + 31 = 36 (x - 4)$

Step 4.2

Apply the distributive property.

$x + 31 = 36 x + 36 \cdot - 4$

Step 4.3

Multiply $36$ by $- 4$ .

$x + 31 = 36 x - 144$

Step 5

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

Tap for more steps...

Step 5.1

Subtract $36 x$ from both sides of the equation.

$x + 31 - 36 x = - 144$

Step 5.2

Subtract $36 x$ from $x$ .

$- 35 x + 31 = - 144$

Step 6

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

Tap for more steps...

Step 6.1

Subtract $31$ from both sides of the equation.

$- 35 x = - 144 - 31$

Step 6.2

Subtract $31$ from $- 144$ .

$- 35 x = - 175$

Step 7

Divide each term in $- 35 x = - 175$ by $- 35$ and simplify.

Tap for more steps...

Step 7.1

Divide each term in $- 35 x = - 175$ by $- 35$ .

$\frac{- 35 x}{- 35} = \frac{- 175}{- 35}$

Step 7.2

Simplify the left side.

Tap for more steps...

Step 7.2.1

Cancel the common factor of $- 35$ .

Tap for more steps...

Step 7.2.1.1

Cancel the common factor.

$\frac{- 35 x}{- 35} = \frac{- 175}{- 35}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 175}{- 35}$

Step 7.3

Simplify the right side.

Tap for more steps...

Step 7.3.1

Divide $- 175$ by $- 35$ .

$x = 5$