Question

Suppose that f (x) is o(g(x)). Does it follow that log |f (x)| is o(log |g(x)|)?

Short answer

Use mathematical representation of little o notation and prove thatit is 0.

Step-by-step solution

Step 1:

Our functions are a(x)=$\log \left|f\right(x\left)\right|$and b(x)=$\log \left|g\right(x\left)\right|$, and we also know that $\underset{x\to \mathrm{\infty }}{lim} \frac{f\left(x\right)}{g\left(x\right)}=0$.

Let our old functions f(x) and g(x) be:

f(x)=$x^2$

g(x)=$x^3$

Then,$\underset{x\to \mathrm{\infty }}{lim} \frac{f\left(x\right)}{g\left(x\right)}=0$

Step 2:

Determining the limit ratios of the new functions:

$\begin{array}{r}\underset{x\to \mathrm{\infty }}{lim} \frac{a\left(x\right)}{b\left(x\right)}=\underset{x\to \mathrm{\infty }}{lim} \frac{\log \left|f\right(x\left)\right|}{\log \left|g\right(x\left)\right|}\\ \Rightarrow \underset{x\to \mathrm{\infty }}{lim} \frac{\log \left|x^2\right|}{\log \left|x^3\right|}=\underset{x\to \mathrm{\infty }}{lim} \frac{\log |x{|}^2}{\log |x{|}^2}=\underset{x\to \mathrm{\infty }}{lim} \frac{2\log \left|x\right|}{3\log \left|x\right|}=\frac{2}{3}\ne 0\end{array}$

We see that a(x) is not o(b(x)).