To analyze and graph the function [tex]\( y = \log_2 x \)[/tex], let's carefully examine its characteristics step-by-step.
1. Domain:
The logarithmic function [tex]\( y = \log_2 x \)[/tex] is defined for all values of [tex]\( x \)[/tex] where [tex]\( x > 0 \)[/tex]. This is because we cannot take the logarithm of zero or a negative number. Therefore, the domain is:
[tex]\[
(0, \infty)
\][/tex]
2. Range:
The range of the logarithmic function [tex]\( y = \log_2 x \)[/tex] consists of all real numbers because the logarithm can produce any real number as output. This is due to the fact that as [tex]\( x \)[/tex] approaches 0 from the positive side, [tex]\( y \)[/tex] approaches negative infinity, and as [tex]\( x \)[/tex] increases beyond 1, [tex]\( y \)[/tex] continues to increase without bound. Therefore, the range is:
[tex]\[
(-\infty, \infty)
\][/tex]
3. Intervals of Increase:
The function [tex]\( y = \log_2 x \)[/tex] is strictly increasing for all [tex]\( x > 0 \)[/tex]. As [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] also increases. Explicitly, for any [tex]\( a \)[/tex] and [tex]\( b \)[/tex] where [tex]\( 0 < a < b \)[/tex], it follows that [tex]\( \log_2 a < \log_2 b \)[/tex]. Therefore, the interval of increase is:
[tex]\[
(0, \infty)
\][/tex]
4. Intervals of Decrease:
The function [tex]\( y = \log_2 x \)[/tex] does not exhibit any intervals of decrease. Since it is strictly increasing for all [tex]\( x > 0 \)[/tex], there are no intervals where the function decreases. Therefore:
[tex]\[
\text{None}
\][/tex]
5. Asymptote(s):
The function [tex]\( y = \log_2 x \)[/tex] has a vertical asymptote at [tex]\( x = 0 \)[/tex]. This is because as [tex]\( x \)[/tex] approaches 0 from the positive side, the logarithm function [tex]\( y \)[/tex] approaches negative infinity, but [tex]\( y \)[/tex] never actually reaches a finite value at [tex]\( x = 0 \)[/tex]. Hence, the asymptote is:
[tex]\[
x = 0
\][/tex]
To summarize:
- Domain: [tex]\( (0, \infty) \)[/tex]
- Range: [tex]\( (-\infty, \infty) \)[/tex]
- Intervals of Increase: [tex]\( (0, \infty) \)[/tex]
- Intervals of Decrease: None
- Asymptote(s): [tex]\( x = 0 \)[/tex]
These characteristics provide a comprehensive understanding of the behavior of the function [tex]\( y = \log_2 x \)[/tex].
