Precalculus Examples

$\log_{3} (\frac{x^{3} (y^{5})}{6})$

Step 1

Rewrite $\log_{3} (\frac{x^{3} (y^{5})}{6})$ as $\log_{3} (x^{3} (y^{5})) - \log_{3} (6)$ .

$\log_{3} (x^{3} (y^{5})) - \log_{3} (6)$

Step 2

Rewrite $\log_{3} (6)$ as $\log_{3} (2 (3))$ .

$\log_{3} (x^{3} (y^{5})) - \log_{3} (2 (3))$

Step 3

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

$\log_{3} (x^{3} (y^{5})) - (\log_{3} (2) + \log_{3} (3))$

Step 4

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

$\log_{3} (x^{3} (y^{5})) - (\log_{3} (2) + 1)$

Step 5

Apply the distributive property.

$\log_{3} (x^{3} (y^{5})) - \log_{3} (2) - 1 \cdot 1$

Step 6

Multiply $- 1$ by $1$ .

$\log_{3} (x^{3} (y^{5})) - \log_{3} (2) - 1$

Step 7

Multiply $x^{3}$ by $y^{5}$ .

$\log_{3} (x^{3} y^{5}) - \log_{3} (2) - 1$

Step 8

Rewrite $\log_{3} (x^{3} y^{5})$ as $\log_{3} (x^{3}) + \log_{3} (y^{5})$ .

$\log_{3} (x^{3}) + \log_{3} (y^{5}) - \log_{3} (2) - 1$

Step 9

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

$3 \log_{3} (x) + \log_{3} (y^{5}) - \log_{3} (2) - 1$

Step 10

Expand $\log_{3} (y^{5})$ by moving $5$ outside the logarithm.

$3 \log_{3} (x) + 5 \log_{3} (y) - \log_{3} (2) - 1$