Precalculus Examples

$4 \log_{3} (x - \frac{1}{3} \cdot \log_{3} (x + 6))$

Step 1

Combine $\log_{3} (x + 6)$ and $\frac{1}{3}$ .

$4 \log_{3} (x - \frac{\log_{3} (x + 6)}{3})$

Step 2

To write $x$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$4 \log_{3} (x \cdot \frac{3}{3} - \frac{\log_{3} (x + 6)}{3})$

Step 3

Combine $x$ and $\frac{3}{3}$ .

$4 \log_{3} (\frac{x \cdot 3}{3} - \frac{\log_{3} (x + 6)}{3})$

Step 4

Combine the numerators over the common denominator.

$4 \log_{3} (\frac{x \cdot 3 - \log_{3} (x + 6)}{3})$

Step 5

Rewrite $\frac{x \cdot 3 - \log_{3} (x + 6)}{3}$ as $(x \cdot 3 - \log_{3} (x + 6)) \cdot 3^{- 1}$ .

$4 \log_{3} ((x \cdot 3 - \log_{3} (x + 6)) \cdot 3^{- 1})$

Step 6

Rewrite $\log_{3} ((x \cdot 3 - \log_{3} (x + 6)) \cdot 3^{- 1})$ as $\log_{3} (x \cdot 3 - \log_{3} (x + 6)) + \log_{3} (3^{- 1})$ .

$4 (\log_{3} (x \cdot 3 - \log_{3} (x + 6)) + \log_{3} (3^{- 1}))$

Step 7

Use logarithm rules to move $- 1$ out of the exponent.

$4 (\log_{3} (x \cdot 3 - \log_{3} (x + 6)) - \log_{3} (3))$

Step 8

Logarithm base $3$ of $3$ is $1$ .

$4 (\log_{3} (x \cdot 3 - \log_{3} (x + 6)) - 1 \cdot 1)$

Step 9

Multiply $- 1$ by $1$ .

$4 (\log_{3} (x \cdot 3 - \log_{3} (x + 6)) - 1)$

Step 10

Move $3$ to the left of $x$ .

$4 (\log_{3} (3 x - \log_{3} (x + 6)) - 1)$

Step 11

Apply the distributive property.

$4 \log_{3} (3 x - \log_{3} (x + 6)) + 4 \cdot - 1$

Step 12

Multiply $4$ by $- 1$ .

$4 \log_{3} (3 x - \log_{3} (x + 6)) - 4$