The following problems are giving me a little grief, mostly because I'm not exactly sure how the solutions should look. Any help is appreciated, I think my biggest problems is not understanding the language of mathematics!

Problem 1Solved

If $\displaystyle \log_{10}2=a$ and $\displaystyle \log_{10}3=b$, find the exact solution to the equation $\displaystyle 12^{x+2}=18^{x-3}$ in terms of $\displaystyle a$ and $\displaystyle b$.

Problem 2Solved

Let $\displaystyle r=\log_{b}\frac{8}{45}$ and $\displaystyle s=\log_{b}\frac{135}{4}$. Find an ordered pair of integers $\displaystyle (m, n)$ such that $\displaystyle \log_{b}\frac{32}{5}=mr+ns$.

Problem 3

Given $\displaystyle \log_{18}6=a$, find $\displaystyle log_{18}16$ in terms of $\displaystyle a$.