# How do you evaluate log_sqrt8 4096?

Nov 17, 2016

${\log}_{\sqrt{8}} 4096 = 8$

#### Explanation:

Let ${\log}_{\sqrt{8}} 4096 = x$

Using definition of logarithm that ${\log}_{a} b = m$ means ${a}^{m} = b$

${\left(\sqrt{8}\right)}^{x} = 4096$

expressing them in powers of $x$

or ${\left({\sqrt{2}}^{3}\right)}^{x} = {2}^{12}$

or ${\left({\left({2}^{\frac{1}{2}}\right)}^{3}\right)}^{x} = {2}^{12}$

or ${2}^{\frac{3 x}{2}} = {2}^{12}$

or $\frac{3 x}{2} = 12$

i.e. $x = 12 \times \frac{2}{3} = 8$

Hence ${\log}_{\sqrt{8}} 4096 = 8$