Answer
(a) Humans are able to perceive approximately 1 octave of visible light.
(b) The microwave region is approximately 13.3 octaves wide.
Work Step by Step
(a) The range of frequencies for visible light is about $4\times 10^{14}~Hz$ to $7.5\times 10^{14}~Hz$. We can find the ratio:
$\frac{7.5\times 10^{14}~Hz}{4\times 10^{14}~Hz} = 1.9$
Since the ratio is 1.9:1, humans are able to perceive approximately 1 octave of visible light.
(b) The range of frequencies for microwaves is about $10^{8}~Hz$ to $10^{12}~Hz$. We can find the ratio:
$\frac{10^{12}~Hz}{10^{8}~Hz} = 10,000$
We can find the number of octaves $N$:
$2^N = 10,000$
$N~ln(2)= ln(10,000)$
$N= \frac{ln(10,000)}{ln(2)}$
$N = 13.3$
The microwave region is approximately 13.3 octaves wide.
