Question Video: Monotonicity of Logarithmic Functions Mathematics

For which values of π‘Ž is the function 𝑓(π‘₯) = log_π‘Ž (π‘₯) decreasing?

01:27

Video Transcript

For which values of π‘Ž is the function 𝑓 of π‘₯ is equal to log base π‘Ž of π‘₯ 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 π‘Ž exists if π‘Ž is greater than zero and not equal to one. We can go one stage further by recalling that if π‘Ž is greater than zero and less than one, 𝑓 of π‘₯ decreases throughout its domain. However, if π‘Ž is greater than one, the function log base π‘Ž of π‘₯ increases throughout its domain.

In this question, we’re asked for the values of π‘Ž for which the function is decreasing. And we can therefore conclude that the function 𝑓 of π‘₯ is equal to log base π‘Ž of π‘₯ is decreasing when π‘Ž 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 π‘Ž of π‘₯, where π‘Ž exists on the open interval between zero and one, is as shown. This confirms that the function is indeed decreasing over its entire domain.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.