What is the degree measure of the least positive angle $x$ for which $\log_2 (\cos x) = −\frac{1}{2}$.

Question

What is the degree measure of the least positive angle $x$ for which $\log_2 (\cos x) = −\frac{1}{2}$.

So i rewrote it as:

$\dfrac{\log (\cos x)}{\log 2}=\dfrac{-1}{2}$

but it doesn't seem to open any doors for me. any ideas?

Answer 1

$ -\frac{1}{2}\log{2} = \log{(1/\sqrt{2})} $. Therefore $$ \log{\cos{x}} = \log{\left(\frac{1}{\sqrt{2}}\right)}. $$ Is it clear now?

Answer 2

Taking $\log x$ to be the natural log,

$$\log (\cos x)=\dfrac{-1}{2}\log 2 $$

$$\log (\cos x)=\log \frac{1}{\sqrt{2}}$$

$$e^{\log (\cos x)}= e^{\log \frac{1}{\sqrt{2}}}$$

Do you see it from here?

Answer 3

Hint: The definition of a logarithm is $$\log_a b=c \iff a^c=b$$

In your case, this means $$\log_2 \cos x =-\frac12 \iff\cos x=\frac{1}{\sqrt{2}}$$

Do you see how to proceed?