logarithmics asymptotics prove(small o)

Question

i was asked to check whether any of those claims is true:

1)$f(n)=o(g(n))$ implies that $\log f(n)=o(\log g(n))$

2)$\log f(n)=o(\log g(n))$ implies that $f(n) = o(g(n))$

what I did:

1) false claim. if $f(n)=o(g(n))$, then it means that $0≤f(n)<cg(n)$, i.e. $\lim _{n\to \infty }\left(\frac{\log f(n)}{\log g(n)}\right)=0$. so if we plug in $f(n)=2^n$ and $g(n)=3^n$, we see that it's not equal to zero and it's a false claim.

2) seems to be true. using the above explanation, $\lim _{n\to \infty }\left(\frac{f(n)}{\log g(n)}\right)=0$, and this seems to be true. couldn't find ay counter example.

Still trying to understand this confusing subject. your input helps me a lot to understand and correct my mistakes.

Thank you very much for your sincere help

Answer 1

Both are not true, for 1) just consider

• $f(n)=\frac 1 {n^2}$
• $g(n)=\frac 1 {n}$

and for 2)

• $ f(n)=e^{\frac 1 {n^2}}$
• $g(n)=e^{\frac 1 {n}}$