# Question Video: Monotonicity of Logarithmic Functions Mathematics

For which values of π is the function π(π₯) = log_π (π₯) decreasing?

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### Video Transcript

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

In this question, weβre asked for the values of π for which the function is decreasing. And we can therefore conclude that the function π of π₯ is equal to log base π of π₯ is decreasing when π exists on the open interval from zero to one.

It is worth noting that the graph of the function π¦ is equal to log base π of π₯, where π exists on the open interval between zero and one, is as shown. This confirms that the function is indeed decreasing over its entire domain.