Precalculus Examples

$\log_{3} (\frac{a \sqrt{b}}{12})$

Step 1

Rewrite $\log_{3} (\frac{a \sqrt{b}}{12})$ as $\log_{3} (a \sqrt{b}) - \log_{3} (12)$ .

$\log_{3} (a \sqrt{b}) - \log_{3} (12)$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{b}$ as $b^{\frac{1}{2}}$ .

$\log_{3} (a b^{\frac{1}{2}}) - \log_{3} (12)$

Step 3

Rewrite $\log_{3} (12)$ as $\log_{3} (2^{2} \cdot 3)$ .

$\log_{3} (a b^{\frac{1}{2}}) - \log_{3} (2^{2} \cdot 3)$

Step 4

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

$\log_{3} (a b^{\frac{1}{2}}) - (\log_{3} (2^{2}) + \log_{3} (3))$

Step 5

Simplify each term.

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Step 5.1

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

$\log_{3} (a b^{\frac{1}{2}}) - (2 \log_{3} (2) + \log_{3} (3))$

Step 5.2

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

$\log_{3} (a b^{\frac{1}{2}}) - (2 \log_{3} (2) + 1)$

Step 6

Apply the distributive property.

$\log_{3} (a b^{\frac{1}{2}}) - (2 \log_{3} (2)) - 1 \cdot 1$

Step 7

Multiply $2$ by $- 1$ .

$\log_{3} (a b^{\frac{1}{2}}) - 2 \log_{3} (2) - 1 \cdot 1$

Step 8

Multiply $- 1$ by $1$ .

$\log_{3} (a b^{\frac{1}{2}}) - 2 \log_{3} (2) - 1$

Step 9

Rewrite $\log_{3} (a b^{\frac{1}{2}})$ as $\log_{3} (a) + \log_{3} (b^{\frac{1}{2}})$ .

$\log_{3} (a) + \log_{3} (b^{\frac{1}{2}}) - 2 \log_{3} (2) - 1$

Step 10

Expand $\log_{3} (b^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{3} (a) + \frac{1}{2} \log_{3} (b) - 2 \log_{3} (2) - 1$

Step 11

Combine $\frac{1}{2}$ and $\log_{3} (b)$ .

$\log_{3} (a) + \frac{\log_{3} (b)}{2} - 2 \log_{3} (2) - 1$