How you solve this? #4^x+9^x+25^x=6^x+10^x+15^x#

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Answer 1 · 2017-03-22T00:06:38

See below.

Calling #X = 2^x, Y = 3^x, Z = 5^x# we have

and making

#vec u = (X,Y,Z)#
#vec v = (X,Z,Y)#
#vec w =(Y,X,Z)#

we have that

#norm(vec u)^2 ge << vec v, vec w >># (Cauchy-Bunyakovsky-Schwarz)

where #<< cdot, cdot >># represents the scalar product of two vectors.

and the equality is verified only for #X=Y=Z#

which implies on

#2^x=3^x=5^x#

This happen for #x=0#