In $2021,$ Ally Bank was offering $7\mathrm{.}2\%$ interest on a CD $($certificate of deposit$),$ with the interest compounded quarterly.

a$.$ Find the associated exponential model for the value of a $$8,000$ investment after $t$ years.

b$.$ Rounding your answer to the nearest dollar, how much would a deposit of $$8,000$ in $2021$ be worth in $2030?$

c$.$ Use logarithms to determine how long $($rounding your answer to the nearest year$)$ it would take for the $$8,000$ investment to be worth $$20,000.$