Precalculus Examples

$\log_{6} ({(\frac{216}{x^{3} y})}^{4})$

Step 1

Expand $\log_{6} ({(\frac{216}{x^{3} y})}^{4})$ by moving $4$ outside the logarithm.

$4 \log_{6} (\frac{216}{x^{3} y})$

Step 2

Rewrite $\log_{6} (\frac{216}{x^{3} y})$ as $\log_{6} (216) - \log_{6} (x^{3} y)$ .

$4 (\log_{6} (216) - \log_{6} (x^{3} y))$

Step 3

Logarithm base $6$ of $216$ is $3$ .

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

Step 4

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

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

Step 5

Expand $\log_{6} (x^{3})$ by moving $3$ outside the logarithm.

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

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $3$ by $- 1$ .

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

Step 7

Apply the distributive property.

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

Step 8

Simplify.

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Step 8.1

Multiply $4$ by $3$ .

$12 + 4 (- 3 \log_{6} (x)) + 4 (- \log_{6} (y))$

Step 8.2

Multiply $- 3$ by $4$ .

$12 - 12 \log_{6} (x) + 4 (- \log_{6} (y))$

Step 8.3

Multiply $- 1$ by $4$ .

$12 - 12 \log_{6} (x) - 4 \log_{6} (y)$