Precalculus Examples

$\log_{a} ({(30 a)}^{3})$

Step 1

Expand $\log_{a} ({(30 a)}^{3})$ by moving $3$ outside the logarithm.

$3 \log_{a} (30 a)$

Step 2

Rewrite $\log_{a} (30 a)$ as $\log_{a} (30) + \log_{a} (a)$ .

$3 (\log_{a} (30) + \log_{a} (a))$

Step 3

Logarithm base $a$ of $a$ is $1$ .

$3 (\log_{a} (30) + 1)$

Step 4

Simplify each term.

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Step 4.1

Rewrite $\log_{a} (30)$ as $\log_{a} (2 \cdot 3 \cdot 5)$ .

$3 (\log_{a} (2 \cdot 3 \cdot 5) + 1)$

Step 4.2

Rewrite $\log_{a} (2 \cdot 3 \cdot 5)$ as $\log_{a} (2 \cdot 3) + \log_{a} (5)$ .

$3 (\log_{a} (2 \cdot 3) + \log_{a} (5) + 1)$

Step 4.3

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

$3 (\log_{a} (2) + \log_{a} (3) + \log_{a} (5) + 1)$

Step 5

Apply the distributive property.

$3 \log_{a} (2) + 3 \log_{a} (3) + 3 \log_{a} (5) + 3 \cdot 1$

Step 6

Multiply $3$ by $1$ .

$3 \log_{a} (2) + 3 \log_{a} (3) + 3 \log_{a} (5) + 3$