Inequalities with logarithms and limits

Question

For my analysis homework, I am to show that $\lim_{n \to \infty} \frac{3^n}{n!} = 0$ using the epsilon definition. My approach is to invoke the squeeze theorem and show that the above sequence is less than $\frac{3^n}{4^n} \forall \hspace{1mm}n\ge 9$; this I can prove via induction. That is, I want to construct the necessary $N$ from $\left(\frac{3}{4}\right)^n < \epsilon$; and so, $n > \log_{\frac{3}{4}}\epsilon$. My question is, why does the inequality switch after I take the logarithm of both sides? Or does it switch at all?

Answer 1

Since the $\ln$ function is monotonically increasing then

$$\left(\frac34\right)^n<\epsilon\iff n\underbrace{\ln\left(\frac34\right)}_{<0}<\ln(\epsilon)\iff n>\frac{\ln(\epsilon)}{\ln\left(\frac34\right)}=\log_{\frac34}(\epsilon)$$

Answer 2

The inequality "switches" (from $<$ to $>$) because you have to effectively divide by a negative number. This is a law of the inequality sign.