Help with radical logs. $\log_ 2\sqrt{32}$

Question

I am having trouble understanding logarithms. Specifically I can't understand this equation.

$$\log_2{\sqrt{32}}$$

I know the answer is $\left(\frac{5}{2}\right)$ but I don't know how it is the answer. If somebody could please explain the process step by step for solving this log and perhaps give examples of other radical logs I would really appreciate it.

Answer 1

Using Logarithmic formula

$$\bullet \log_{e}(m)^n = n\cdot \log_{e}(m)\;,$$ where $m>0$ and $$ \bullet \log_{m}(m) =1\;,$$ Where $m>0$ and $m\neq 1$

So $$\displaystyle \log_{2}\sqrt{32} = \log_{2}(32)^{\frac{1}{2}} =\log_{2}\left(2^5\right)^{\frac{1}{2}}= \log_{2}(2)^{\frac{5}{2}} = \frac{5}{2}\log_{2}(2) = \frac{5}{2}$$

Answer 2

Hint : $$\large \sqrt{32}=32^{\frac{1}{2}}=2^{\frac{5}{2}}$$