Expand the Logarithmic Expression log of ( square root of a^3b^5)/(c^7)

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

Step 1

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

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a^{3} b^{5}}$ as ${(a^{3} b^{5})}^{\frac{1}{2}}$ .

$\log ({(a^{3} b^{5})}^{\frac{1}{2}}) - \log (c^{7})$

Step 3

Expand $\log ({(a^{3} b^{5})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (a^{3} b^{5}) - \log (c^{7})$

Step 4

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

$\frac{1}{2} \log (a^{3} b^{5}) - (7 \log (c))$

Step 5

Multiply $7$ by $- 1$ .

$\frac{1}{2} \log (a^{3} b^{5}) - 7 \log (c)$

Step 6

Rewrite $\log (a^{3} b^{5})$ as $\log (a^{3}) + \log (b^{5})$ .

$\frac{1}{2} (\log (a^{3}) + \log (b^{5})) - 7 \log (c)$

Step 7

Expand $\log (a^{3})$ by moving $3$ outside the logarithm.

$\frac{1}{2} (3 \log (a) + \log (b^{5})) - 7 \log (c)$

Step 8

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

$\frac{1}{2} (3 \log (a) + 5 \log (b)) - 7 \log (c)$

Step 9

Simplify each term.

Apply the distributive property.

$\frac{1}{2} (3 \log (a)) + \frac{1}{2} (5 \log (b)) - 7 \log (c)$

Multiply $\frac{1}{2} (3 \log (a))$ .

Combine $3$ and $\frac{1}{2}$ .

$\frac{3}{2} \log (a) + \frac{1}{2} (5 \log (b)) - 7 \log (c)$

Combine $\frac{3}{2}$ and $\log (a)$ .

$\frac{3 \log (a)}{2} + \frac{1}{2} (5 \log (b)) - 7 \log (c)$

Multiply $\frac{1}{2} (5 \log (b))$ .

Combine $5$ and $\frac{1}{2}$ .

$\frac{3 \log (a)}{2} + \frac{5}{2} \log (b) - 7 \log (c)$

Combine $\frac{5}{2}$ and $\log (b)$ .

$\frac{3 \log (a)}{2} + \frac{5 \log (b)}{2} - 7 \log (c)$