Expand the Logarithmic Expression log of square root of c^5d

$\log (\sqrt{c^{5} d})$

Step 1

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

$\log ({(c^{5} d)}^{\frac{1}{2}})$

Step 2

Expand $\log ({(c^{5} d)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (c^{5} d)$

Step 3

Rewrite $\log (c^{5} d)$ as $\log (c^{5}) + \log (d)$ .

$\frac{1}{2} (\log (c^{5}) + \log (d))$

Step 4

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

$\frac{1}{2} (5 \log (c) + \log (d))$

Step 5

Apply the distributive property.

$\frac{1}{2} (5 \log (c)) + \frac{1}{2} \log (d)$

Step 6

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

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

$\frac{5}{2} \log (c) + \frac{1}{2} \log (d)$

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

$\frac{5 \log (c)}{2} + \frac{1}{2} \log (d)$

Step 7

Combine $\frac{1}{2}$ and $\log (d)$ .

$\frac{5 \log (c)}{2} + \frac{\log (d)}{2}$