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

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

Step 1

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

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

Step 2

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

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

Step 3

Rewrite $\log (\frac{2 x}{y})$ as $\log (2 x) - \log (y)$ .

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

Step 4

Rewrite $\log (2 x)$ as $\log (2) + \log (x)$ .

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

Step 5

Apply the distributive property.

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

Step 6

Simplify.

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

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

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

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

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

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