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}}})

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

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

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

\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}}

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

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

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

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

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

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

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

Step 3

Expand

\log (a^{6})

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

6

$6$ outside the logarithm.

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

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

Step 4

Expand

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

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

\frac{1}{2}

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

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

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

Step 5

Combine

\frac{1}{2}

$\frac{1}{2}$ and

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

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

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

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

Step 6

Rewrite

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

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

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

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

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

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

Step 7

Expand

\log (b^{3})

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

3

$3$ outside the logarithm.

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

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

Step 8

Expand

\log (c^{3})

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

3

$3$ outside the logarithm.

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

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

Step 9

Factor

3

$3$ out of

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

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

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Step 9.1

Factor

3

$3$ out of

3 \log (b)

$3 \log (b)$ .

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

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

Step 9.2

Factor

3

$3$ out of

3 \log (c)

$3 \log (c)$ .

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

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

Step 9.3

Factor

3

$3$ out of

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

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

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

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

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

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

Step 10

To write

6 \log (a)

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

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 11

Combine

6 \log (a)

$6 \log (a)$ and

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 13

Simplify the numerator.

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Step 13.1

Factor

3

$3$ out of

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

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

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Step 13.1.1

Factor

3

$3$ out of

6 \log (a) \cdot 2

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

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

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

Step 13.1.2

Factor

3

$3$ out of

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

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

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

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

Step 13.1.3

Factor

3

$3$ out of

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

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

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

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

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

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

Step 13.2

Multiply

2

$2$ by

2

$2$ .

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

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

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

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