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Which properties are present in a table that represents a logarithmic function in the form [tex]\(y=\log_8 x\)[/tex] when [tex]\(b\ \textgreater \ 1\)[/tex]?

I. The [tex]\(y\)[/tex]-values are always increasing or always decreasing.
II. The point [tex]\((1,0)\)[/tex] exists in the table.
III. The [tex]\(y\)[/tex]-values will decrease rapidly as the [tex]\(x\)[/tex]-values approach zero.
IV. There will only be one [tex]\(x\)[/tex]-value in the table with a [tex]\(y\)[/tex]-value of zero.

A. I only
B. I and II only
C. I, III, and IV
D. II and III only

Sagot :

GinnyAnsweridn
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.
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