Question

If f(x) = loga(2574), a > 0, a ≠ 1, then f(x) > 0 when:

A. a > 1
B. a < 1
C. a = 1
D. a ≠ 1

Answer 1

The function f(x) = loga(2574) will be greater than zero if the base a is greater than 1, which corresponds to answer choice A (a > 1).

Step-by-step explanation:

The question asks us to determine for what values of a, where a > 0 and a ≠ 1, the function f(x) = loga(2574) will be greater than zero (f(x) > 0). To answer this, we look at the properties of logarithms and the behavior of log functions.

For any base a that is greater than 1, the logarithmic function is increasing. This means that as x gets larger, loga(x) will also become larger. Given that 2574 is a number greater than 1, if a is chosen to be greater than 1 as well, loga(2574) will be positive because we are, in a sense, counting how many times we must multiply a by itself to get 2574, and since 2574 is a substantial number, this will be a positive count.

On the other hand, if a is between 0 and 1, the logarithmic function is decreasing, which implies that loga(x) for x > 1 will yield a negative value. Therefore, in order for f(x) to be greater than 0, the base a must be greater than 1. So the correct answer is A. a > 1.