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

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify each term.

Apply the distributive property.

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

Cancel the common factor of $2$ .

Factor $2$ out of $2 \log (x)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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