Precalculus Examples

$2 \log_{3} (\frac{6 x^{2}}{4 y^{3}})$

Step 1

Rewrite $\log_{3} (\frac{6 x^{2}}{4 y^{3}})$ as $\log_{3} (6 x^{2}) - \log_{3} (4 y^{3})$ .

$2 (\log_{3} (6 x^{2}) - \log_{3} (4 y^{3}))$

Step 2

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

$2 (\log_{3} (6) + \log_{3} (x^{2}) - \log_{3} (4 y^{3}))$

Step 3

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

$2 (\log_{3} (6) + 2 \log_{3} (x) - \log_{3} (4 y^{3}))$

Step 4

Rewrite $\log_{3} (4 y^{3})$ as $\log_{3} (4) + \log_{3} (y^{3})$ .

$2 (\log_{3} (6) + 2 \log_{3} (x) - (\log_{3} (4) + \log_{3} (y^{3})))$

Step 5

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

$2 (\log_{3} (6) + 2 \log_{3} (x) - (\log_{3} (4) + 3 \log_{3} (y)))$

Step 6

Simplify each term.

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Step 6.1

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

$2 (\log_{3} (2 (3)) + 2 \log_{3} (x) - (\log_{3} (4) + 3 \log_{3} (y)))$

Step 6.2

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

$2 (\log_{3} (2) + \log_{3} (3) + 2 \log_{3} (x) - (\log_{3} (4) + 3 \log_{3} (y)))$

Step 6.3

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

$2 (\log_{3} (2) + 1 + 2 \log_{3} (x) - (\log_{3} (4) + 3 \log_{3} (y)))$

Step 6.4

Simplify each term.

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Step 6.4.1

Rewrite $\log_{3} (4)$ as $\log_{3} (2^{2})$ .

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

Step 6.4.2

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

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

Step 6.5

Apply the distributive property.

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

Step 6.6

Multiply $2$ by $- 1$ .

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

Step 6.7

Multiply $3$ by $- 1$ .

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

Step 7

Subtract $2 \log_{3} (2)$ from $\log_{3} (2)$ .

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

Step 8

Apply the distributive property.

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

Step 9

Simplify.

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Step 9.1

Multiply $2$ by $1$ .

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

Step 9.2

Multiply $2$ by $2$ .

$2 + 4 \log_{3} (x) + 2 (- \log_{3} (2)) + 2 (- 3 \log_{3} (y))$

Step 9.3

Multiply $- 1$ by $2$ .

$2 + 4 \log_{3} (x) - 2 \log_{3} (2) + 2 (- 3 \log_{3} (y))$

Step 9.4

Multiply $- 3$ by $2$ .

$2 + 4 \log_{3} (x) - 2 \log_{3} (2) - 6 \log_{3} (y)$