All exponential functions can be written in the form #y=ab^x#. We just need to solve for #a# and #b#.
Since the function goes through the points #(1,6)# and #(2,12)#, #x=1=>y=6# and #x=2=>y=12\ \ \ \ (1)#.
Since #x=1=>y=6#, we just need to input these values into the equation #y=ab^x# to get #6=ab#.
Since #x=2=>y=12#, we just need to input these values into the equation #y=ab^x# to get #12=ab^2\ \ \ \ (2)#.
Since neither #a# nor #b# can be zero, we can safely divide equation #(2)# by equation #(1)# to get #12/6=(ab^2)/(ab)#, or #2=b#. Since #6=ab#, it must be the case that #a=3#.
Our final exponential function is #y=3*2^x#.
We can rewrite this function in several other forms, such as #y=3*2^x=2^ln(3)*2^x=2^(x+ln(3))# or #y=3*2^x=3*e^(xln(2))=3*10^(xlog_10(2))#.
