Precalculus Examples

$\log_{3} (9 x^{4}) - \log_{3} ({(3 x)}^{2})$

Step 1

Rewrite $\log_{3} (9 x^{4})$ as $\log_{3} (9) + \log_{3} (x^{4})$ .

$\log_{3} (9) + \log_{3} (x^{4}) - \log_{3} ({(3 x)}^{2})$

Step 2

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

$\log_{3} (9) + 4 \log_{3} (x) - \log_{3} ({(3 x)}^{2})$

Step 3

Logarithm base $3$ of $9$ is $2$ .

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Multiply $2$ by $1$ .

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

Step 7.3

Apply the distributive property.

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

Step 7.4

Multiply $- 1$ by $2$ .

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

Step 7.5

Multiply $2$ by $- 1$ .

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

Step 8

Subtract $2$ from $2$ .

$4 \log_{3} (x) + 0 - 2 \log_{3} (x)$

Step 9

Add $4 \log_{3} (x)$ and $0$ .

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

Step 10

Subtract $2 \log_{3} (x)$ from $4 \log_{3} (x)$ .

$2 \log_{3} (x)$