# How do you calculate log_11 (1/sqrt 11)?

Sep 8, 2016

${\log}_{11} \left(\frac{1}{\sqrt{11}}\right) = - \frac{1}{2}$

#### Explanation:

The question being asked in this log form is..

${\log}_{11} \left(\frac{1}{\sqrt{11}}\right)$

"What index/power of 11 will give $\frac{1}{\sqrt{11}}$?"

The answer is not immediately obvious, so let's use some of the laws of indices to change $\frac{1}{\sqrt{11}}$ into a different form.

$\frac{1}{\sqrt{11}} = \frac{1}{11} ^ \left(\frac{1}{2}\right) = {11}^{-} \left(\frac{1}{2}\right)$

${\log}_{11} {11}^{- \frac{1}{2}}$ is now obvious by inspection,
because we can see that 11 has the index $- \frac{1}{2}$

${\log}_{11} \left(\frac{1}{\sqrt{11}}\right) = - \frac{1}{2}$

We could also get to the same result by using the change of base law.

${\log}_{11} \left(\frac{1}{\sqrt{11}}\right) = \frac{\log \left(\frac{1}{\sqrt{11}}\right)}{\log} 11 = - \frac{1}{2}$