By inspection x=0 is a solution.
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.

Posted by Jer
on 20210605 10:27:40 