# How do you solve for k in (1/2)^(10/k) = 0.8?

May 26, 2016

$k = 31.06$

#### Explanation:

${\left(\frac{1}{2}\right)}^{\frac{10}{k}} = 0.8 = \frac{4}{5}$. Now taking log on both sides

$\left(\frac{10}{k}\right) \log \left(\frac{1}{2}\right) = \log \left(\frac{4}{5}\right)$ or

$\left(\frac{10}{k}\right) = \log \frac{\frac{4}{5}}{\log} \left(\frac{1}{2}\right)$ or

$10 = k \cdot \log \frac{\frac{4}{5}}{\log} \left(\frac{1}{2}\right)$ or

$k = \frac{10}{\log \frac{\frac{4}{5}}{\log} \left(\frac{1}{2}\right)}$ or

$k = 10 \cdot \log \frac{\frac{1}{2}}{\log} \left(\frac{4}{5}\right) = 10 \cdot \frac{- 0.3010}{- 0.09691}$

or $k = 31.06$