Any logarithm with a base that is greater than 1 has a graph that is always increasing.
Conversely, when b is between 0 and 1, the logarithm's graph is decreasing. This is because the base is then a decimal number and whenever we raise a decimal number to a power, we get an even smaller number. Let's draw the given graph.
So far we know that the logarithm's base falls somewhere between 0 and 1. To narrow down the interval, we will add the graphs of two logarithmic functions. One that goes through (2,-1) and another that goes through (2,-2).
Using the known points that fall on the line, we can write two equations. One describes the red curve (I) and the other describes the green curve (II).
(I):& log_b 2 = -1
(II):& log_b 2 = -2
If we use the definition of a logarithm, we can rewrite these into exponential form.
(I):& b^(-1)= 2
(II):& b^(-2)= 2
Let's solve for b in these equations.
