Let f(x)=2^x + 3^x + 6^x

g(x)= 4^x + 9^x

then the equation becomes f(x)-g(x)=1

These functions both have domain of the reals and range of the positive reals.  Also, both are strictly increasing over the reals.

Since f(-1)=1 there can be no solutions < -1

Since f(1)=11 and g(1)=13 there can be no solutions > 1

A quick look at the graph in this region shows there is indeed a single solution.  f(x)-g(x) reaches a maximum at (0,1).  This could easily be proven by finding the first and second derivatives if we needed to satisfy our Calc I teacher.