Expand the Logarithmic Expression log of square root of 10000x

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Apply the distributive property.

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

Step 6

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 7

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

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