$5.$ Suppose that a sum $\backslash ($ S$\_\{0\}\backslash )$ is invested at an annual rate of return $\backslash ($ r $\backslash )$ compounded continuously.

a$.$ Find the time $\backslash ($ T $\backslash )$ required for the original sum to double in value as a function of $\backslash ($ r $\backslash ).$

b$.$ Determine $\backslash ($ T $\backslash )$ if $\backslash ($ r$=7\backslash \%\backslash ).$

c$.$ Find the return rate that must be achieved if the initial investment is to double in $8$ years.