Some algebraic step I'm missing? Exponentiation and log related. $100(1-\frac{12 \cdot 0.85^{a+1}}{12 \cdot 0.85^a})$  
As seen here(Or pic below), this evaluates to 15.  

There is one step I am worried is wrong, can I just rewrite the exponentiations over like this?  
$100(1-\frac{12 \cdot (a+1) \cdot \log 0.85}{12 \cdot a \cdot \log 0.85})$ 
Because in the end I end up with the following
$\frac{100}{a}$
Instead of 15. 

No, you took the logarithm of one piece of the expression, which will not preserve anything about it (and note, even if you took the logarithm of the entire expression, you would then be looking to prove it was equal to $\log(15)$, not $15$). For example, if we had the expression
$$1-\frac{10^5}{10^2}$$
(which is equal to $1-1000=-999$)
it's incorrect to claim it's equal to 
$$1-\frac{5\log(10)}{2\log(10)}=1-\frac{5}{2}=-\frac{3}{2}$$
However, in your problem, what you can do is note that
$$0.85^{a+1}=0.85\times 0.85^a$$
and divide.
