Expand the Logarithmic Expression log of ( square root of z^3)/(x^2y^5)

$\log (\frac{\sqrt{z^{3}}}{x^{2} y^{5}})$

Step 1

Rewrite $\log (\frac{\sqrt{z^{3}}}{x^{2} y^{5}})$ as $\log (\sqrt{z^{3}}) - \log (x^{2} y^{5})$ .

$\log (\sqrt{z^{3}}) - \log (x^{2} y^{5})$

Step 2

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

$\log (z^{\frac{3}{2}}) - \log (x^{2} y^{5})$

Step 3

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

$\frac{3}{2} \log (z) - \log (x^{2} y^{5})$

Step 4

Rewrite $\log (x^{2} y^{5})$ as $\log (x^{2}) + \log (y^{5})$ .

$\frac{3}{2} \log (z) - (\log (x^{2}) + \log (y^{5}))$

Step 5

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

$\frac{3}{2} \log (z) - (2 \log (x) + \log (y^{5}))$

Step 6

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

$\frac{3}{2} \log (z) - (2 \log (x) + 5 \log (y))$

Step 7

Simplify each term.

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

$\frac{3 \log (z)}{2} - (2 \log (x) + 5 \log (y))$

Apply the distributive property.

$\frac{3 \log (z)}{2} - (2 \log (x)) - (5 \log (y))$

Multiply $2$ by $- 1$ .

$\frac{3 \log (z)}{2} - 2 \log (x) - (5 \log (y))$

Multiply $5$ by $- 1$ .

$\frac{3 \log (z)}{2} - 2 \log (x) - 5 \log (y)$