For which values of 𝑎 is the function 𝑓 of 𝑥 is equal to log base 𝑎 of 𝑥 decreasing?
This question can be answered directly from our definition of the logarithmic function. We recall that a logarithmic function is the inverse of an exponential function. And log base 𝑎 exists if 𝑎 is greater than zero and not equal to one. We can go one stage further by recalling that if 𝑎 is greater than zero and less than one, 𝑓 of 𝑥 decreases throughout its domain. However, if 𝑎 is greater than one, the function log base 𝑎 of 𝑥 increases throughout its domain.
In this question, we’re asked for the values of 𝑎 for which the function is decreasing. And we can therefore conclude that the function 𝑓 of 𝑥 is equal to log base 𝑎 of 𝑥 is decreasing when 𝑎 exists on the open interval from zero to one.
It is worth noting that the graph of the function 𝑦 is equal to log base 𝑎 of 𝑥, where 𝑎 exists on the open interval between zero and one, is as shown. This confirms that the function is indeed decreasing over its entire domain.