Consider y = log_b a. Match the following:

List-I | List-II
P. If a > 1 and b > 1, then | 1. y > 0
Q. If a > 1 and 0 < b < 1, then | 2. y < 0
R. If 0 < a < 1 and 0 < b < 1, then | 3. y ≥ 1
S. If 0 < a < 1 and b > 1, then | 4. y ≤ -1
| 5. y = 0

Options:
(A) P-3, Q-4, R-5, S-4
(B) P-3, Q-4, R-1, S-2
(C) P-1, Q-2, R-1, S-2
(D) P-1, Q-5, R-3, S-4

To solve this problem, we need to understand the behavior of logarithms and how the base and the argument affect the sign and value of the logarithmic expression.
Let's consider the expression y = lo g b ​ a . This expresses the power to which the base b must be raised to obtain the number a .
Let's analyze each scenario:
List-I | List-II
P: If 1"> a > 1 and 1"> b > 1 , then:

0"> lo g b ​ a > 0 .

This is because for a base greater than 1 and an argument greater than 1, the logarithm is positive. The base must be raised to a power greater than 0 to reach a .
Q: If 1"> a > 1 and 0 < b < 1 , then:

lo g b ​ a < 0 .

Here, the base is a fraction and must be raised to a negative power to produce a larger number a .
R: If 0 < a < 1 and 0 < b < 1 , then:

0"> lo g b ​ a > 0 .

Both the base and the argument are fractions. The base, being a fraction, would require a positive power to become smaller to match the argument a .
S: If 0 < a < 1 and 1"> b > 1 , then:

lo g b ​ a < 0 .

The base b is greater than 1, and a is a fraction, thus the power must be negative to make a greater base reduce to a number smaller than 1.
Therefore, the correct matching option of List-I to List-II is:
(A). P-1, Q-2, R-1, S-2

Answered by OliviaMariThompson | 2025-07-06