For what values of b will F(x) = log(base b)x be an increasing function?

Question

b > 1? b < 1? b > 0? Or, b < 0?

Answer 1

any b >1

there are severe problems talking about logs to a negative base
if y = log[-2] (2) it would mean that (-2)^y = 2

similarly y = log[1]x would mean 1^y = x

for 0< b<1
y= log[b]x would mean b^y = 1/(1/b)^y = x so (1/b)^y = 1/x
and decreasing

Answer 2

b > 1

Answer 3

At b between 1 and zero, the function decreases, since you are multiplying some decimal over and over again.
At zero, the function will be constant and zero all the time.
At less than zero, the function will rise and fall depending on x.
At exactly one the function becomes F(x) = x and again, whether it rises or falls depends on x.

Only at b > 1 will the function rise.

Answer 4

log(base b) x = ln(x)/ln(b)
so F'(x) = 1/x *1/lnb>0 so ln b >0 and b>1