How do you evaluate $log_4 (1/2)$ ?

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Answer 1 · 2016-01-07T20:00:30

$log_4(1/2) = -1/2$

Using the following properties:

we have

$log_4(1/2) = log_4(4^(-1/2))$

$= -1/2log_4(4)$

$= -1/2(1)$

$= -1/2$

Answer 2 · 2016-01-07T20:03:34

$log_4(1/2)=(-1/2)$

Based on the meaning of $log$ and exponents:
If
$color(white)("XXX")log_4(1/2) = c$
then
$color(white)("XXX")4^c= 1/2$

We know that
$color(white)("XXX")4^c>=1$ for $c>=0$
So
$color(white)("XXX")4^c=1/2 rarr c<0$

Also
$color(white)("XXX")4^((-k)) = 1/(4^k)$

Which leads us to asking for what value of $k$ does
$color(white)("XXX")1/(4^k) = 1/2$ ?

Since
$color(white)("XXX")4^(1/2) = sqrt(4) = 2$

$1/2 = 1/(4^(1/2)) = 4^(-1/2)$
and
$color(white)("XXX")c=(-1/2)$

How do you evaluate log_4 (1/2)log_4 (1/2)?