How do you use the Change of Base Formula and a calculator to evaluate the logarithm log_2 sqrt2?

Sep 2, 2015

Try this:

Explanation:

Let us see the formula:
${\log}_{b} x = {\log}_{\textcolor{red}{c}} \frac{x}{\log} _ \textcolor{red}{c} b$
So let us choose a convenient log such as the natural logarithm $\ln$ ($= {\log}_{e}$) that can be easily evaluated by our pocket calculator and apply the formula to get:
${\log}_{2} \sqrt{2} = {\log}_{e} \frac{\sqrt{2}}{\log} _ e 2 = \ln \frac{\sqrt{2}}{\ln} 2 = 0.5$

Another way (maybe easier) is to consider:
$\sqrt{2} = {2}^{\frac{1}{2}}$
and
$\log {x}^{a} = a \log x$
so you get:
${\log}_{2} \sqrt{2} = {\log}_{2} {\left(2\right)}^{\frac{1}{2}} = \frac{1}{2} {\log}_{2} \left(2\right) = \frac{1}{2} \times 1 = \frac{1}{2}$