# How do you simplify log_4 8?

May 7, 2016

Use the logarithmic properties:

${\log}_{a} \left({b}^{c}\right) = c \cdot {\log}_{a} \left(b\right)$

${\log}_{a} \left(b\right) = {\log}_{c} \frac{b}{\log} _ c \left(a\right)$

You can notice that $c = 2$ fits this case since $8$ can be derived as a power of $2$. Answer is:

${\log}_{4} 8 = 1.5$

#### Explanation:

${\log}_{4} 8$

${\log}_{2} \frac{8}{\log} _ \left(2\right) 4$

${\log}_{2} {2}^{3} / {\log}_{2} {2}^{2}$

$\frac{3 \cdot {\log}_{2} 2}{2 \cdot {\log}_{2} 2}$

$\frac{3}{2}$

$1.5$