Expand the Logarithmic Expression log base b of square root of 2x

$\log_{b} (\sqrt{2 x})$

Step 1

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

$\log_{b} ({(2 x)}^{\frac{1}{2}})$

Step 2

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

$\frac{1}{2} \log_{b} (2 x)$

Step 3

Rewrite $\log_{b} (2 x)$ as $\log_{b} (2) + \log_{b} (x)$ .

$\frac{1}{2} (\log_{b} (2) + \log_{b} (x))$

Step 4

Apply the distributive property.

$\frac{1}{2} \log_{b} (2) + \frac{1}{2} \log_{b} (x)$

Step 5

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

$\frac{\log_{b} (2)}{2} + \frac{1}{2} \log_{b} (x)$

Step 6

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

$\frac{\log_{b} (2)}{2} + \frac{\log_{b} (x)}{2}$