Question

Which properties are present in a table that represents a logarithmic function in the form mc027-1.jpgwhen mc027-2.jpg?
I. The y-values are always increasing or always decreasing.
II. The point (0, 1) exists in the table.
III. The y-values will decrease rapidly as the x-values approach zero.
IV. There will only be one x-value in the table with a y-value of zero.
I only
I and II only
I, III, and IV
II and III only

Answer 1

Answer: I, III, and IV.

Explanation:
- I: True — for a fixed base a>1, y = log_a(x) is monotonic (always increasing); for 0<a<1 it would be always decreasing, so in either case the y-values are monotonic.
- II: False — log_a(0) is undefined, and (0,1) is not on a logarithm. (The point (0,1) is on an exponential y = a^x, not a log.)
- III: True (for a>1) — as x → 0+, log_a(x) → −∞, so the y-values decrease without bound.
- IV: True — log_a(x) = 0 only when x = 1, so there is exactly one x-value with y = 0.