试证函数F(x,y) = ln x ⋅ ln y满足关系式 F(xy,uv) = F(x,u) + F(x,v) + F( y,u) + F( y,v) .
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![试证函数F(x,y) = ln x ⋅ ln y满足关系式 F(xy,uv) = F(x,u) + F(x,v) + F( y,u) + F( y,v) .](/uploads/image/z/9292816-64-6.jpg?t=%E8%AF%95%E8%AF%81%E5%87%BD%E6%95%B0F%28x%2Cy%29+%3D+ln+x+%26%238901%3B+ln+y%E6%BB%A1%E8%B6%B3%E5%85%B3%E7%B3%BB%E5%BC%8F+F%28xy%2Cuv%29+%3D+F%28x%2Cu%29+%2B+F%28x%2Cv%29+%2B+F%28+y%2Cu%29+%2B+F%28+y%2Cv%29+.)
试证函数F(x,y) = ln x ⋅ ln y满足关系式 F(xy,uv) = F(x,u) + F(x,v) + F( y,u) + F( y,v) .
试证函数F(x,y) = ln x ⋅ ln y满足关系式 F(xy,uv) = F(x,u) + F(x,v) + F( y,u) + F( y,v) .
试证函数F(x,y) = ln x ⋅ ln y满足关系式 F(xy,uv) = F(x,u) + F(x,v) + F( y,u) + F( y,v) .
证
F(xy,uv) = ln(xy) ⋅ ln(uv) = (ln x + ln y) ⋅ (ln u + ln v)
= ln x ⋅ ln u + ln x ⋅ ln v + ln y ⋅ ln u + ln y ⋅ ln v
= F(x,u) + F(x,v) + F( y,u) + F( y,v) .