Trouble in calculating which function grows faster I have a trouble in calculating which function grows faster. 
$f(n) = 3\log_4 n + \sqrt{n} + 3 \\
g(n) = 4\log_3 n + \log n + 200$
Can someone let me know how to solve this? 
 A: $\log n$ grows slower than any $n^d$, in particular $\sqrt{n}$.
Then, look at the terms which grow the fastest in $f$ and $g$.
It is clear that $f$ will grow faster because it has the $\sqrt{n}$ term.
You can show this concretely by considering 
$$\lim_{n\to\infty} \frac{f(n)}{g(n)}$$
and showing it goes to $\infty$.
edit:
$$\begin{align}
\lim_{n\to\infty} \frac{f(n)}{g(n)} &= \lim_{n\to\infty}\frac{3\log_4 n + \sqrt{n} + 3}{4\log_3 n + \log n + 200}
\\ &= \lim_{n\to\infty} \frac{1/\sqrt{n}}{1/\sqrt{n}}\frac{3\log_4 n + \sqrt{n} + 3}{4\log_3 n + \log n + 200}
\\&=\lim_{n\to\infty} \frac{3\log_4 n/\sqrt{n} + 1 + 3/\sqrt{n}}{4\log_3 n /\sqrt{n}+ \log n/\sqrt{n} + 200/\sqrt{n}}
\\&\to \infty
\end{align}$$
since $\log n/\sqrt{n}\to 0$ (for any log base).
A: You can ignore the constants at the end because they don't effect the growth rate of the functions. If you then compare e to the power of f and g, you will see that e^g is of the form cn^2, while e^f is of the form cne^sqrt(n). Can you tell which of these grows faster?
