To determine the properties of a table that represents a logarithmic function [tex]\( y = \log_b x \)[/tex] when [tex]\( b > 1 \)[/tex], we need to evaluate each property one by one.
Property I:
The [tex]\( y \)[/tex]-values are always increasing or always decreasing.
For [tex]\( y = \log_b x \)[/tex] with [tex]\( b > 1 \)[/tex], the logarithmic function is increasing. As [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] also increases. Therefore, the [tex]\( y \)[/tex]-values are always increasing.
Conclusion for Property I: True
Property II:
The point [tex]\( (0, 1) \)[/tex] exists in the table.
For the function [tex]\( y = \log_b x \)[/tex], it is important to note that the logarithmic function is undefined at [tex]\( x = 0 \)[/tex]. Thus, the point [tex]\( (0, 1) \)[/tex] cannot exist in the table because the logarithm function does not take [tex]\( x = 0 \)[/tex].
Conclusion for Property II: False
Property III:
The [tex]\( y \)[/tex]-values will decrease rapidly as the [tex]\( x \)[/tex]-values approach zero.
As [tex]\( x \)[/tex] approaches zero from the positive side, the value of [tex]\( y = \log_b x \)[/tex] decreases towards negative infinity. Therefore, the [tex]\( y \)[/tex]-values decrease rapidly as [tex]\( x \)[/tex] approaches zero.
Conclusion for Property III: True
Property IV:
There will only be one [tex]\( x \)[/tex]-value in the table with a [tex]\( y \)[/tex]-value of zero.
For the function [tex]\( y = \log_b x \)[/tex], [tex]\( y \)[/tex] equals zero when [tex]\( x \)[/tex] equals 1 because [tex]\( \log_b 1 = 0 \)[/tex]. There is only one such x-value where this occurs (which is [tex]\( x = 1 \)[/tex]).
Conclusion for Property IV: True
Combining the conclusions from above:
- Property I: True
- Property II: False
- Property III: True
- Property IV: True
The properties present in the table are I, III, and IV. Therefore, the correct option is:
I, III, and IV
Hence, the correct answer is:
3) I, III, and IV.
