若3^15a=5^5b=15^3c,则5ab-bc-3ac=?

先假设3^15a=5^5b=15^3c=x
对各等式取以x为底的对数，
logx 3^15a=logx 5^5b=logx 15^3c=logx x
15a*logx 3=5b*logx 5=3c*logx 15=1
a=1/(15*logx 3)
b=1/(5*logx 5)
c=1/(3*logx 15)
5ab-bc-3ac=5*1/(15*logx 3)*1/(5*logx 5)-1/(5*logx 5)*1/(3*logx 15)-3*1/(15*logx 3)*1/(3*logx 15)=1/15(1/(logx 3*logx 5)-1/(logx 5*logx 15)-1/(logx 3*logx 15))
=1/15(1/(logx 3*logx 5)-(logx 3+logx 5)/(logx 3*logx 5*logx 15))
=1/15(1/(logx 3*logx 5)-logx 15/(logx 3*logx 5*logx 15))
=1/15(1/(logx 3*logx 5)-1/(logx 3*logx 5))
=0

ln(3^15a)=ln(5^5b)=ln(15^3c)
15aln(3)=5bln(5)=3cln(15)
b=3alog5(3)
c=5alog15(3)
5ab-bc-3ac=15a^2log5(3)-15a^2log5(3)log15(3)-15a^2log15(3)
=15a^2[log5(3)-log5(3)log15(3)-log15(3)]
=15a^2[ln(3)/ln(5)-ln(3)ln(3)/ln(5)ln(15)-ln(3)/ln(15)]
=15a^2ln3[ln(15)-ln(3)-ln(5)]/[ln(15)ln(15)]
因为ln(15)=ln(3)+ln(5)
所以原式=0