Expand the Logarithmic Expression log of square root of (xy^7)/(z^3)

$\log (\sqrt{\frac{x y^{7}}{z^{3}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{x y^{7}}{z^{3}}}$ as ${(\frac{x y^{7}}{z^{3}})}^{\frac{1}{2}}$ .

$\log ({(\frac{x y^{7}}{z^{3}})}^{\frac{1}{2}})$

Step 2

Expand $\log ({(\frac{x y^{7}}{z^{3}})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (\frac{x y^{7}}{z^{3}})$

Step 3

Rewrite $\log (\frac{x y^{7}}{z^{3}})$ as $\log (x y^{7}) - \log (z^{3})$ .

$\frac{1}{2} (\log (x y^{7}) - \log (z^{3}))$

Step 4

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

$\frac{1}{2} (\log (x y^{7}) - (3 \log (z)))$

Step 5

Multiply $3$ by $- 1$ .

$\frac{1}{2} (\log (x y^{7}) - 3 \log (z))$

Step 6

Rewrite $\log (x y^{7})$ as $\log (x) + \log (y^{7})$ .

$\frac{1}{2} (\log (x) + \log (y^{7}) - 3 \log (z))$

Step 7

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

$\frac{1}{2} (\log (x) + 7 \log (y) - 3 \log (z))$

Step 8

Apply the distributive property.

$\frac{1}{2} \log (x) + \frac{1}{2} (7 \log (y)) + \frac{1}{2} (- 3 \log (z))$

Step 9

Simplify.

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

$\frac{\log (x)}{2} + \frac{1}{2} (7 \log (y)) + \frac{1}{2} (- 3 \log (z))$

Multiply $\frac{1}{2} (7 \log (y))$ .

Combine $7$ and $\frac{1}{2}$ .

$\frac{\log (x)}{2} + \frac{7}{2} \log (y) + \frac{1}{2} (- 3 \log (z))$

Combine $\frac{7}{2}$ and $\log (y)$ .

$\frac{\log (x)}{2} + \frac{7 \log (y)}{2} + \frac{1}{2} (- 3 \log (z))$

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

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

$\frac{\log (x)}{2} + \frac{7 \log (y)}{2} + \frac{- 3}{2} \log (z)$

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

$\frac{\log (x)}{2} + \frac{7 \log (y)}{2} + \frac{- 3 \log (z)}{2}$

Step 10

Move the negative in front of the fraction.

$\frac{\log (x)}{2} + \frac{7 \log (y)}{2} - \frac{3 \log (z)}{2}$