Hey guys, I was just wondering if any of you have any thoughts on this problem. I've exhausted all mine

Let's say we have 2 functions: f and g with $\displaystyle f(x)=3^x$ and $\displaystyle g(x)=100^x$.

Then define two sequences:

1) $\displaystyle a_1$ = 3 and $\displaystyle a_{n+1}$ = $\displaystyle f(a_n)$ for n $\displaystyle \geq$ 1

2) $\displaystyle b_1$ = 100 and $\displaystyle b_{n+1}$ = $\displaystyle g(b_n)$ for n $\displaystyle \geq$ 1

What is the smallest positive integer m for which $\displaystyle b_m > a_{100}$?

I've tried connecting the two series with an inequality but I can't quite get one that I can prove. I've probably overlooked something blindingly obvious

Thanks in advance if you guys can help