which properties are present in a table that represents a logarithmic function in the form ( y = log_{b}x )…

which properties are present in a table that represents a logarithmic function in the form ( y = log_{b}x ) when ( b > 1 )?\ni. the ( y )-values are always increasing or always decreasing.\nii. the point ( (0, 1) ) exists in the table.\niii. the ( y )-values will decrease rapidly as the ( x )-values approach zero.\niv. there will only be one ( x )-value in the table with a ( y )-value of zero.\no i only\no i and ii only\no i, iii, and iv\no ii and iii only

which properties are present in a table that represents a logarithmic function in the form ( y = log_{b}x ) when ( b > 1 )?\ni. the ( y )-values are always increasing or always decreasing.\nii. the point ( (0, 1) ) exists in the table.\niii. the ( y )-values will decrease rapidly as the ( x )-values approach zero.\niv. there will only be one ( x )-value in the table with a ( y )-value of zero.\no i only\no i and ii only\no i, iii, and iv\no ii and iii only

Answer

Brief Explanations:

  • For property I:
    • The logarithmic function (y = \log_{b}x) with (b>1) is a one - to - one function. Its derivative is (y^\prime=\frac{1}{x\ln b}), and since (x>0) (domain of (y = \log_{b}x)) and (b > 1) (so (\ln b>0)), the function is increasing for all (x>0). So the (y) - values are always increasing.
  • For property II:
    • Substitute (x = 0) into (y=\log_{b}x). The function (y = \log_{b}x) is not defined at (x = 0) because (\lim_{x\rightarrow0^{+}}\log_{b}x=-\infty). The point ((0,1)) does not lie on the graph of (y=\log_{b}x).
  • For property III:
    • As (x\rightarrow0^{+}), using the limit (\lim_{x\rightarrow0^{+}}\log_{b}x=-\infty) (since (b > 1)). So the (y) - values will decrease rapidly as (x) - values approach zero.
  • For property IV:
    • Set (y = 0), then (0=\log_{b}x). By the definition of logarithms ((y=\log_{b}x) if and only if (x = b^{y})), when (y = 0), (x=b^{0}=1). So there is only one (x) (i.e., (x = 1)) for which (y = 0) on the graph of (y=\log_{b}x).

Answer:

C. I, III, and IV