Expand the Logarithmic Expression log of square root of 10000z

$\log (\sqrt{10000 z})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10000 z}$ as ${(10000 z)}^{\frac{1}{2}}$ .

$\log ({(10000 z)}^{\frac{1}{2}})$

Expand $\log ({(10000 z)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (10000 z)$

Rewrite $\log (10000 z)$ as $\log (10000) + \log (z)$ .

$\frac{1}{2} (\log (10000) + \log (z))$

Logarithm base $10$ of $10000$ is $4$ .

$\frac{1}{2} (4 + \log (z))$

Apply the distributive property.

$\frac{1}{2} \cdot 4 + \frac{1}{2} \log (z)$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$\frac{1}{2} \cdot (2 (2)) + \frac{1}{2} \log (z)$

Cancel the common factor.

$\frac{1}{2} \cdot (2 \cdot 2) + \frac{1}{2} \log (z)$

Rewrite the expression.

$2 + \frac{1}{2} \log (z)$

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

$2 + \frac{\log (z)}{2}$