Expand the Logarithmic Expression log base 3 of x/12

$\log_{3} (\frac{x}{12})$

Step 1

Rewrite $\log_{3} (\frac{x}{12})$ as $\log_{3} (x) - \log_{3} (12)$ .

$\log_{3} (x) - \log_{3} (12)$

Step 2

Rewrite $\log_{3} (12)$ as $\log_{3} (2^{2} \cdot 3)$ .

$\log_{3} (x) - \log_{3} (2^{2} \cdot 3)$

Step 3

Rewrite $\log_{3} (2^{2} \cdot 3)$ as $\log_{3} (2^{2}) + \log_{3} (3)$ .

$\log_{3} (x) - (\log_{3} (2^{2}) + \log_{3} (3))$

Step 4

Simplify each term.

Expand $\log_{3} (2^{2})$ by moving $2$ outside the logarithm.

$\log_{3} (x) - (2 \log_{3} (2) + \log_{3} (3))$

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

$\log_{3} (x) - (2 \log_{3} (2) + 1)$

Step 5

Apply the distributive property.

$\log_{3} (x) - (2 \log_{3} (2)) - 1 \cdot 1$

Step 6

Multiply $2$ by $- 1$ .

$\log_{3} (x) - 2 \log_{3} (2) - 1 \cdot 1$

Step 7

Multiply $- 1$ by $1$ .

$\log_{3} (x) - 2 \log_{3} (2) - 1$