Expand the Logarithmic Expression log of ( square root of zy^3)/x

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

Step 1

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

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

Step 2

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

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

Step 3

Rewrite $\log (z^{\frac{1}{2}} y^{3})$ as $\log (z^{\frac{1}{2}}) + \log (y^{3})$ .

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

Step 4

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

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

Step 5

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

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

Step 6

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

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