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

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify each term.

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

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

Apply the distributive property.

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

Multiply $2$ by $- 1$ .

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