Expand the Logarithmic Expression log base 2 of fourth root of c^10d^16

$\log_{2} (\sqrt[4]{c^{10} d^{16}})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{c^{10} d^{16}}$ as ${(c^{10} d^{16})}^{\frac{1}{4}}$ .

$\log_{2} ({(c^{10} d^{16})}^{\frac{1}{4}})$

Expand $\log_{2} ({(c^{10} d^{16})}^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\frac{1}{4} \log_{2} (c^{10} d^{16})$

Rewrite $\log_{2} (c^{10} d^{16})$ as $\log_{2} (c^{10}) + \log_{2} (d^{16})$ .

$\frac{1}{4} (\log_{2} (c^{10}) + \log_{2} (d^{16}))$

Expand $\log_{2} (c^{10})$ by moving $10$ outside the logarithm.

$\frac{1}{4} (10 \log_{2} (c) + \log_{2} (d^{16}))$

Expand $\log_{2} (d^{16})$ by moving $16$ outside the logarithm.

$\frac{1}{4} (10 \log_{2} (c) + 16 \log_{2} (d))$

Apply the distributive property.

$\frac{1}{4} (10 \log_{2} (c)) + \frac{1}{4} (16 \log_{2} (d))$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$\frac{1}{2 (2)} (10 \log_{2} (c)) + \frac{1}{4} (16 \log_{2} (d))$

Factor $2$ out of $10 \log_{2} (c)$ .

$\frac{1}{2 (2)} (2 (5 \log_{2} (c))) + \frac{1}{4} (16 \log_{2} (d))$

Cancel the common factor.

$\frac{1}{2 \cdot 2} (2 (5 \log_{2} (c))) + \frac{1}{4} (16 \log_{2} (d))$

Rewrite the expression.

$\frac{1}{2} (5 \log_{2} (c)) + \frac{1}{4} (16 \log_{2} (d))$

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

$\frac{5}{2} \log_{2} (c) + \frac{1}{4} (16 \log_{2} (d))$

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

$\frac{5 \log_{2} (c)}{2} + \frac{1}{4} (16 \log_{2} (d))$

Cancel the common factor of $4$ .

Factor $4$ out of $16 \log_{2} (d)$ .

$\frac{5 \log_{2} (c)}{2} + \frac{1}{4} (4 (4 \log_{2} (d)))$

Cancel the common factor.

$\frac{5 \log_{2} (c)}{2} + \frac{1}{4} (4 (4 \log_{2} (d)))$

Rewrite the expression.

$\frac{5 \log_{2} (c)}{2} + 4 \log_{2} (d)$