For what values of b will F(x) = logb x be an increasing function? A. b < 0 B. b > 0 C. b < 1 D. b > 1,

In order to answer you need to keep in mind what is a logarithm:
logb x = n is the equivalent of asking "you should raise the base b to what number n in order to get x?": bⁿ = x

From this follows that b must be positive, otherwise the function is not defined everywhere in R: for example, take log(-2) x = 0.5 ? This means: [tex] (-2)^{1/2} [/tex] =√(-2) which is impossible in R. 
Therefore, both answers A and C are not correct because they take in consideration negative values.

If we take the base between 0 and 1 excluded, we have a fraction whose denominator is bigger than the numerator. Such a fraction, elevated to increasing exponents get smaller, for example:
(1/2)² = 1/4
(1/2)³ = 1/8
(1/2)⁴ = 1/16
Therefore the function is dereasing and answer B is incorrect.

The right answer is, therefore, D) b > 1.