Precalculus Examples

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

Step 1

Apply the product rule to $30 a$ .

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

Step 2

Rewrite $\log_{a} (30^{3} a^{3})$ as $\log_{a} (30^{3}) + \log_{a} (a^{3})$ .

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

Step 3

Use logarithm rules to move $3$ out of the exponent.

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

Step 4

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

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

Step 5

Multiply $3$ by $1$ .

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

Step 6

Raise $30$ to the power of $3$ .

$\log_{a} (27000) + 3$

Step 7

Rewrite $\log_{a} (27000)$ using the change of base formula.

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Step 7.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} + 3$

Step 7.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (27000)}{\log (a)} + 3$