Algebra Examples

$\log_{6} (\frac{11 c^{- 9}}{d \sqrt[3]{6 - d}})$

Step 1

Rewrite $\log_{6} (\frac{11 c^{- 9}}{d \sqrt[3]{6 - d}})$ as $\log_{6} (11 c^{- 9}) - \log_{6} (d \sqrt[3]{6 - d})$ .

$\log_{6} (11 c^{- 9}) - \log_{6} (d \sqrt[3]{6 - d})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{6 - d}$ as ${(6 - d)}^{\frac{1}{3}}$ .

$\log_{6} (11 c^{- 9}) - \log_{6} (d {(6 - d)}^{\frac{1}{3}})$

Step 3

Rewrite $\log_{6} (11 c^{- 9})$ as $\log_{6} (11) + \log_{6} (c^{- 9})$ .

$\log_{6} (11) + \log_{6} (c^{- 9}) - \log_{6} (d {(6 - d)}^{\frac{1}{3}})$

Step 4

Expand $\log_{6} (c^{- 9})$ by moving $- 9$ outside the logarithm.

$\log_{6} (11) - 9 \log_{6} (c) - \log_{6} (d {(6 - d)}^{\frac{1}{3}})$

Step 5

Rewrite $\log_{6} (d {(6 - d)}^{\frac{1}{3}})$ as $\log_{6} (d) + \log_{6} ({(6 - d)}^{\frac{1}{3}})$ .

$\log_{6} (11) - 9 \log_{6} (c) - (\log_{6} (d) + \log_{6} ({(6 - d)}^{\frac{1}{3}}))$

Step 6

Expand $\log_{6} ({(6 - d)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\log_{6} (11) - 9 \log_{6} (c) - (\log_{6} (d) + \frac{1}{3} \log_{6} (6 - d))$

Step 7

Simplify each term.

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Step 7.1

Combine $\frac{1}{3}$ and $\log_{6} (6 - d)$ .

$\log_{6} (11) - 9 \log_{6} (c) - (\log_{6} (d) + \frac{\log_{6} (6 - d)}{3})$

Step 7.2

To write $\log_{6} (d)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\log_{6} (11) - 9 \log_{6} (c) - (\log_{6} (d) \cdot \frac{3}{3} + \frac{\log_{6} (6 - d)}{3})$

Step 7.3

Combine $\log_{6} (d)$ and $\frac{3}{3}$ .

$\log_{6} (11) - 9 \log_{6} (c) - (\frac{\log_{6} (d) \cdot 3}{3} + \frac{\log_{6} (6 - d)}{3})$

Step 7.4

Combine the numerators over the common denominator.

$\log_{6} (11) - 9 \log_{6} (c) - \frac{\log_{6} (d) \cdot 3 + \log_{6} (6 - d)}{3}$

Step 7.5

Move $3$ to the left of $\log_{6} (d)$ .

$\log_{6} (11) - 9 \log_{6} (c) - \frac{3 \log_{6} (d) + \log_{6} (6 - d)}{3}$