Expand the Logarithmic Expression log of (c^7)/( cube root of a^7b^5)

$\log (\frac{c^{7}}{\sqrt[3]{a^{7} b^{5}}})$

Rewrite $\log (\frac{c^{7}}{\sqrt[3]{a^{7} b^{5}}})$ as $\log (c^{7}) - \log (\sqrt[3]{a^{7} b^{5}})$ .

$\log (c^{7}) - \log (\sqrt[3]{a^{7} b^{5}})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{a^{7} b^{5}}$ as ${(a^{7} b^{5})}^{\frac{1}{3}}$ .

$\log (c^{7}) - \log ({(a^{7} b^{5})}^{\frac{1}{3}})$

Expand $\log (c^{7})$ by moving $7$ outside the logarithm.

$7 \log (c) - \log ({(a^{7} b^{5})}^{\frac{1}{3}})$

Expand $\log ({(a^{7} b^{5})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$7 \log (c) - (\frac{1}{3} \log (a^{7} b^{5}))$

Combine $\frac{1}{3}$ and $\log (a^{7} b^{5})$ .

$7 \log (c) - \frac{\log (a^{7} b^{5})}{3}$

Rewrite $\log (a^{7} b^{5})$ as $\log (a^{7}) + \log (b^{5})$ .

$7 \log (c) - \frac{\log (a^{7}) + \log (b^{5})}{3}$

Expand $\log (a^{7})$ by moving $7$ outside the logarithm.

$7 \log (c) - \frac{7 \log (a) + \log (b^{5})}{3}$

Expand $\log (b^{5})$ by moving $5$ outside the logarithm.

$7 \log (c) - \frac{7 \log (a) + 5 \log (b)}{3}$

To write $7 \log (c)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$7 \log (c) \cdot \frac{3}{3} - \frac{7 \log (a) + 5 \log (b)}{3}$

Combine $7 \log (c)$ and $\frac{3}{3}$ .

$\frac{7 \log (c) \cdot 3}{3} - \frac{7 \log (a) + 5 \log (b)}{3}$

Combine the numerators over the common denominator.

$\frac{7 \log (c) \cdot 3 - (7 \log (a) + 5 \log (b))}{3}$

Simplify the numerator.

Multiply $3$ by $7$ .

$\frac{21 \log (c) - (7 \log (a) + 5 \log (b))}{3}$

Apply the distributive property.

$\frac{21 \log (c) - (7 \log (a)) - (5 \log (b))}{3}$

Multiply $7$ by $- 1$ .

$\frac{21 \log (c) - 7 \log (a) - (5 \log (b))}{3}$

Multiply $5$ by $- 1$ .

$\frac{21 \log (c) - 7 \log (a) - 5 \log (b)}{3}$