已知3^15a=5^5b=15^3c,求证：5ab-bc-3ac=0
1个回答
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
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