4^(x+1.5) + 9^(x+0.5) = 10×6^x             (1)


It is equivalent to


2^(2x+3) + 3^(2x+1) = 10*(2^x)*(3^x),   or


8*2^(2x) + 3*3^(2x) = 10*(2^x)*(3^x).      (2) 


Introduce new variables  u = 2^x,  v = 3^x.  Then equation (2) takes the form


    8u^2 + 3v^2 = 10uv


Divide both sides by v^2.  You will get


     + 3 -  = 0                (3)


Let z = .   Then equation (3) takes the form


    8z^2 - 10z + 3 = 0.


Solve this quadratic equation using the quadratic formula


     =  = .


The roots are  


     =  =  = ,   and

     =  =  = .


Thus, we should consider two cases.


(a)   = .


     It means   =  = .


     Next you can take any logarithm, log base 10, or natural logarithm "ln" to continue


        = ,

        = 

        x =  =  = 0.709511  (approximately).

The first case is completed.


Similarly, you can consider and complete the second case   = .


May I leave it to you, in order for you completed it on your own ?

As an accurate person, I checked my answer.


I used MS Excel in my computer and calculated the left side and the right side expressions of the original equation at x= 0.709511.


I got the value of  35.65389  on the left side and the value of  35.65389  on the right side.