Expand the Logarithmic Expression log of square root of 1000000z

$\log (\sqrt{1000000 z})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1000000 z}$ as ${(1000000 z)}^{\frac{1}{2}}$ .

$\log ({(1000000 z)}^{\frac{1}{2}})$

Step 2

Expand $\log ({(1000000 z)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (1000000 z)$

Step 3

Rewrite $\log (1000000 z)$ as $\log (1000000) + \log (z)$ .

$\frac{1}{2} (\log (1000000) + \log (z))$

Step 4

Logarithm base $10$ of $1000000$ is $6$ .

$\frac{1}{2} (6 + \log (z))$

Step 5

Apply the distributive property.

$\frac{1}{2} \cdot 6 + \frac{1}{2} \log (z)$

Step 6

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

$\frac{1}{2} \cdot (2 (3)) + \frac{1}{2} \log (z)$

Cancel the common factor.

$\frac{1}{2} \cdot (2 \cdot 3) + \frac{1}{2} \log (z)$

Rewrite the expression.

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

Step 7

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

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