# 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? – Atul Ingle Feb 12 '17 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. – Ankit Sahay Feb 12 '17 at 16:19
• This is a memoryless system. It is static. In that sense, it is internally stable. – Rodrigo de Azevedo Feb 13 '17 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.