Algebra Examples

log (\frac{a^{6}}{\sqrt{b^{3} c^{3}}})

$\log (\frac{a^{6}}{\sqrt{b^{3} c^{3}}})$

Step 1

Rewrite

$\log (\frac{a^{6}}{\sqrt{b^{3} c^{3}}})$ as

$\log (a^{6}) - \log (\sqrt{b^{3} c^{3}})$ .

$\log (a^{6}) - \log (\sqrt{b^{3} c^{3}})$

Step 2

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{b^{3} c^{3}}$ as

${(b^{3} c^{3})}^{\frac{1}{2}}$ .

$\log (a^{6}) - \log ({(b^{3} c^{3})}^{\frac{1}{2}})$

Step 3

Expand

$\log (a^{6})$ by moving

$6$ outside the logarithm.

$6 \log (a) - \log ({(b^{3} c^{3})}^{\frac{1}{2}})$

Step 4

Expand

$\log ({(b^{3} c^{3})}^{\frac{1}{2}})$ by moving

$\frac{1}{2}$ outside the logarithm.

$6 \log (a) - (\frac{1}{2} \log (b^{3} c^{3}))$

Step 5

Combine

$\frac{1}{2}$ and

$\log (b^{3} c^{3})$ .

$6 \log (a) - \frac{\log (b^{3} c^{3})}{2}$

Step 6

Rewrite

$\log (b^{3} c^{3})$ as

$\log (b^{3}) + \log (c^{3})$ .

$6 \log (a) - \frac{\log (b^{3}) + \log (c^{3})}{2}$

Step 7

Expand

$\log (b^{3})$ by moving

$3$ outside the logarithm.

$6 \log (a) - \frac{3 \log (b) + \log (c^{3})}{2}$

Step 8

Expand

$\log (c^{3})$ by moving

$3$ outside the logarithm.

$6 \log (a) - \frac{3 \log (b) + 3 \log (c)}{2}$

Step 9

Factor

$3$ out of

$3 \log (b) + 3 \log (c)$ .

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Step 9.1

Factor

$3$ out of

$3 \log (b)$ .

$6 \log (a) - \frac{3 (\log (b)) + 3 \log (c)}{2}$

Step 9.2

Factor

$3$ out of

$3 \log (c)$ .

$6 \log (a) - \frac{3 (\log (b)) + 3 (\log (c))}{2}$

Step 9.3

Factor

$3$ out of

$3 (\log (b)) + 3 (\log (c))$ .

$6 \log (a) - \frac{3 (\log (b) + \log (c))}{2}$

Step 10

To write

$6 \log (a)$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$6 \log (a) \cdot \frac{2}{2} - \frac{3 (\log (b) + \log (c))}{2}$

Step 11

Combine

$6 \log (a)$ and

$\frac{2}{2}$ .

$\frac{6 \log (a) \cdot 2}{2} - \frac{3 (\log (b) + \log (c))}{2}$

Step 12

Combine the numerators over the common denominator.

$\frac{6 \log (a) \cdot 2 - 3 (\log (b) + \log (c))}{2}$

Step 13

Simplify the numerator.

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Step 13.1

Factor

$3$ out of

$6 \log (a) \cdot 2 - 3 (\log (b) + \log (c))$ .

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Step 13.1.1

Factor

$3$ out of

$6 \log (a) \cdot 2$ .

$\frac{3 (2 \log (a) \cdot 2) - 3 (\log (b) + \log (c))}{2}$

Step 13.1.2

Factor

$3$ out of

$- 3 (\log (b) + \log (c))$ .

$\frac{3 (2 \log (a) \cdot 2) + 3 (- (\log (b) + \log (c)))}{2}$

Step 13.1.3

Factor

$3$ out of

$3 (2 \log (a) \cdot 2) + 3 (- (\log (b) + \log (c)))$ .

$\frac{3 (2 \log (a) \cdot 2 - (\log (b) + \log (c)))}{2}$

Step 13.2

Multiply

$2$ by

$2$ .

$\frac{3 (4 \log (a) - (\log (b) + \log (c)))}{2}$

Step 13.3

Apply the distributive property.

$\frac{3 (4 \log (a) - \log (b) - \log (c))}{2}$