# How do you find the exact value of log_2 root4 8?

##### 1 Answer
Oct 1, 2017

$\frac{3}{4}$

#### Explanation:

$\text{using the "color(blue)"law of logarithms}$

•color(white)(x)log_bx=nhArrx=b^n

$\Rightarrow {\log}_{2} \left(\sqrt[4]{8}\right) = n$

$\Rightarrow \sqrt[4]{8} = {2}^{n}$

$\text{using the "color(blue)"law of exponents}$

•color(white)(x)a^(m/n)=root(n)(a^m)

$\Rightarrow \sqrt[4]{{2}^{3}} = {2}^{\frac{3}{4}} = {2}^{n}$

$\text{since the bases on both sides are 2 we can equate}$
$\text{the exponents}$

$\Rightarrow n = \frac{3}{4}$

$\Rightarrow {\log}_{2} \sqrt[4]{8} = \frac{3}{4}$