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

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

Step 1

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

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z^{5}}$ as $z^{\frac{5}{2}}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Rewrite $\log (x^{3} y^{2})$ as $\log (x^{3}) + \log (y^{2})$ .

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

Step 6

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

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

Step 7

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

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