solve 3 * 4^x - 6^x = 2 * 9^x

we have 4^x = 2 ^ 2x
6^x = 2^x 3^ x
9^x = 3^2x

let 2^x = a and 3^x = b ( we have a and b are not independent of each other) ..1

3a^2 - ab = 2b^2

or 3a^2 - ab - 2b^2 = 0

or (3a + 2b)(a-b) = 0

a = b or (a/b) =1 = (2/3)^x ( from 1) so x = 0

or 3a = - 2b which is not possible as a and b both >0 from (1)

so x = 0