Question

已知3^15a=5^5b=15^3c,求证：5ab-bc-3ac=0

Answer 1

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