jmc

We can rewrite $5^b+5^b+5^b+5^b+5^b$ as $5\cdot 5^b=5^{(b+1)}$. Since $625=5^4$, we rewrite $625^{(b-1)}$ as $(5^4{)}^{(b-1)}=5^{4(b-1)}=5^{(4b-4)}$. We now have $5^{(b+1)}=5^{(4b-4)}$, so it must be that the exponents are equal. $$b+1=4b-4\Rightarrow 5=3b\Rightarrow \frac{5}{3}=b$$ The value of $b$ is $\boxed{\frac{5}{3}}$.