Expand the Logarithmic Expression log base 5 of 8( square root of t)/v

$\log_{5} (8 \frac{\sqrt{t}}{v})$

Rewrite $\log_{5} (8 \frac{\sqrt{t}}{v})$ as $\log_{5} (8) + \log_{5} (\frac{\sqrt{t}}{v})$ .

$\log_{5} (8) + \log_{5} (\frac{\sqrt{t}}{v})$

Rewrite $\log_{5} (\frac{\sqrt{t}}{v})$ as $\log_{5} (\sqrt{t}) - \log_{5} (v)$ .

$\log_{5} (8) + \log_{5} (\sqrt{t}) - \log_{5} (v)$

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

$\log_{5} (8) + \log_{5} (t^{\frac{1}{2}}) - \log_{5} (v)$

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

$\log_{5} (8) + \frac{1}{2} \log_{5} (t) - \log_{5} (v)$

Simplify each term.

Rewrite $\log_{5} (8)$ as $\log_{5} (2^{3})$ .

$\log_{5} (2^{3}) + \frac{1}{2} \log_{5} (t) - \log_{5} (v)$

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

$3 \log_{5} (2) + \frac{1}{2} \log_{5} (t) - \log_{5} (v)$

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

$3 \log_{5} (2) + \frac{\log_{5} (t)}{2} - \log_{5} (v)$