# Find $x = \log_2(4^{3\over 4}\cdot \sqrt{2^5})^{1 \over 2}$

I am looking for $x$.

$$x = \log_2 \left[(4^{3\over 4}\cdot \sqrt{2^5})^{1 \over 2}\right]$$

I am not sure how to do this.

I am trying to solve this by changing the form like this:

$$x = \log_a b \Rightarrow a^x =b$$

thanks

$$x = \log_2(4^{3\over 4}\cdot \sqrt{2^5})^{1 \over 2}=\frac{1}{2}\log_2(2^{3/2}\times 2^{5/2})=\frac{1}{2}\log_2 2^4=\frac{4}{2}\log_2 2 =2\ .$$
Use sum rule to get $x=1/2(log_2(4^{3/4}.2^{5/2}))$ $=1/2(log_2(4^{3/4})+log_2(2^{5/2}))$ $=1/2(3/4(log_2(4))+5/2(log_2(2)))$ $=1/2(3/4(log_2(2^2))+5/2(log_2(2)))$ $=1/2(3/4(2log_2(2))+5/2(log_2(2)))$ $=1/2(3/2(log_2(2))+5/2(log_2(2)))$ $=1/2(4(log_2(2)))$ $=2(log_2(2))$ $=2(1)$ $=2$