ozlem,
\(\displaystyle what \ if \ you \ let \ log(n) = x^2 \ ?\)
\(\displaystyle Then \ 10^{log(n)} \ = \ 10^{x^2}\)
\(\displaystyle So \ n \ = \ 10^{x^2}\)
\(\displaystyle Suppose \ x \ = \ \sqrt{log(n)}\)
\(\displaystyle As \ n \ \longrightarrow \infty, \ log(n) \ \longrightarrow \infty. \ And \ then \ \sqrt{log(n)} \longrightarrow \infty.\)
\(\displaystyle So \ then \ x \ \longrightarrow \infty.\)
As an example, look at transforming your second problem:
\(\displaystyle \lim_{x \to \infty} \frac{2^x}{10^{x^2}} \ = \ \lim_{x \to \infty} \frac{2^x}{10^{x(x)}} \ = \ \lim_{x \to \infty} \frac{(2)^x}{(10^x)^x} \ = \ \lim_{x \to \infty}\bigg(\frac{2}{10^x}\bigg)^x\)
What can you show/state after this?
