# Stability of a logarithmic function and system

A system such that

$$y[n] = \log_{10}(|x[n]|)$$

was given as a stable one. I am not able to understand it though. When $x[n] = 0$ (which is bounded), the output $y[n]$ tends to get unbounded. So how is this system stable?

• Where "was it given"? I agree with your reasoning that it's not stable. Perhaps they are using a more restrictive definition of stability? Feb 12, 2017 at 15:40
• The question belongs from a book named "Signals and Systems" by Simon Haykin, although the solution does not come from a reliable source. Thank you for clearing the confusion. Feb 12, 2017 at 16:19
• This is a memoryless system. It is static. In that sense, it is internally stable. Feb 13, 2017 at 1:02

As correctly pointed out in a comment, this system is definitely not BIBO-stable. The output signal becomes arbitrarily large (in magnitude) when $x(t)$ approaches zero. So a bounded input signal can result in an unbounded output signal, and, consequently, the system is unstable.
Suppose there is an M$$\in R$$ such that: $$|x(n)| now you need to show that there is an $$L \in R$$ such that $$|y[n]| < L \forall n \in N$$
This is not a BIBO -stable system . Simple counterexample: $$x(t) = |sint| \rightarrow |x(t) | <2$$ but $$\lim_{t->0}|y(t)| = \lim_{t\rightarrow 0}|log(|sint|)|=\infty$$
side note : $$y(t)$$ is well defined for $$: t \neq k\pi$$