# logarithmic derivative

It seems that when the logarithmic derivative of a function exists, the function itself cannot be zero. But I failed to construct a convincent proof. Is this true?

$\dfrac{f'(x)}{f(x)}$ is bounded implies $f(x)$ cannot have zero as a limiting value.

Is this true?

-
For $f'(x)/f(x)$ to even make sense, you need $f(x)\neq 0$. – 1015 Mar 4 '13 at 18:51
Also, for $\ln f(x)$ to even make sense, you need $f(x)\ne 0$. – Hagen von Eitzen Mar 4 '13 at 18:59
@HagenvonEitzen I think you meant $f(x)>0$. – 1015 Mar 4 '13 at 19:04
Thanks, I was not precise enough. Of course f(x)≠0. But can f(x) have zero as a limiting value? – will Mar 4 '13 at 20:23
I think this is kind of 0/0 limit... – will Mar 4 '13 at 20:39

Let $f:x\to \frac{1}{x}$ on $[1,+\infty)$, then we have $$\frac{f'(x)}{f(x)}=-\frac{1}{x},$$ so $\displaystyle\frac{f'}{f}$ is bounded on $[1,+\infty)$ Nevertheless $\displaystyle \lim_{+\infty}f=0.$

-
The statement is true at finite points. – 1015 Mar 4 '13 at 20:50
Okay, thanks +1 – user63181 Mar 4 '13 at 21:05

True when $x$ approaches a finite $x_0$, not when $x$ approaches $\pm \infty$, as explained below.

Assume $f$ is differentiable on $\mathbb{R}$.

Then the function $$g(x):=\ln|f(x)|$$ is defined and differentiable on the open set $$U=\{x\in\mathbb{R}\;;\; f(x)\neq 0\}.$$ Its derivative is $$g'(x)=\frac{f'(x)}{f(x)}\qquad \forall\;x\in U.$$

Write $U=\bigcup_{n\geq 1}(a_n,b_n)$ as a countable disjoint union of open intervals. See this thread to see why this is possible.

Consider for instance a finite point $x_0=a_n$.

Note that $\lim_{x\rightarrow x_0^+}\ln |f(x)|=\lim_{x\rightarrow x_0^+}g(x)=-\infty$.

Now if $g'=(\ln |f|)'$ was bounded by, say, $M$ on $(x_0,x_0+\delta]$, the mean value theorem, and then the triangular inequality, would yield $$|g(x)-g(x_0+\delta)|\leq M \delta\quad \Rightarrow\quad |g(x)|\leq |g(x_0+\delta)|+M\delta$$ for all $x\in(x_0,x_0+\delta]$.

So the logarithmic derivative is unbounded when it approaches every finite boundary point of $U$.
Now regarding the behavior at $\pm\infty$, you can easily find examples where $f'/f$ is bounded while $f$ tends to $0$.
I'm not sure I understood the $\Rightarrow$ properly. I get how the left-hand side is true but I'm not sure I understand the implication. Could you please add some more details? – xavierm02 Mar 4 '13 at 21:12
@xavierm02 Ah, ok. This is the triangular inequality: $|g(x)|=|g(x)-g(y)+g(y)|\leq |g(x)-g(y)|+|g(y)|$. – 1015 Mar 4 '13 at 21:32
@xavierm02 No, I use the fact that every open set in $\mathbb{R}$ is the disjoint union of countably many open intervals. – 1015 Mar 4 '13 at 21:42