Expand the Logarithmic Expression log base 3 of ( square root of 5)/(q^2p)

$\log_{3} (\frac{\sqrt{5}}{q^{2} p})$

Rewrite $\log_{3} (\frac{\sqrt{5}}{q^{2} p})$ as $\log_{3} (\sqrt{5}) - \log_{3} (q^{2} p)$ .

$\log_{3} (\sqrt{5}) - \log_{3} (q^{2} p)$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\log_{3} (5^{\frac{1}{2}}) - \log_{3} (q^{2} p)$

Expand $\log_{3} (5^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{3} (5) - \log_{3} (q^{2} p)$

Combine $\frac{1}{2}$ and $\log_{3} (5)$ .

$\frac{\log_{3} (5)}{2} - \log_{3} (q^{2} p)$

Rewrite $\log_{3} (q^{2} p)$ as $\log_{3} (q^{2}) + \log_{3} (p)$ .

$\frac{\log_{3} (5)}{2} - (\log_{3} (q^{2}) + \log_{3} (p))$

Expand $\log_{3} (q^{2})$ by moving $2$ outside the logarithm.

$\frac{\log_{3} (5)}{2} - (2 \log_{3} (q) + \log_{3} (p))$

Simplify each term.

Apply the distributive property.

$\frac{\log_{3} (5)}{2} - (2 \log_{3} (q)) - \log_{3} (p)$

Multiply $2$ by $- 1$ .

$\frac{\log_{3} (5)}{2} - 2 \log_{3} (q) - \log_{3} (p)$