设log(a)（x+y)=根号三,log(a)x=1,求log(a)y

题目

设log(a)（x+y)=根号三,log(a)x=1,求log(a)y

答案

∵log(a)（x+y)=√3,∴x+y=a^√3.
∵log(a)x=1,∴x=a^1=a.y=(x+y)-x=(a^√3)-a=a(a^(√3-1)-1).
两边取对数,log(a)y=log(a)a(a^(√3-1)-1)=log(a)a+log(a)(a^(√3-1)-1)=1+log(a)(a^(√3-1)-1).
举例说明一下：假设a=10,则x=a^1=a=10.
log(a)（x+y)=√3=log(10)（x+y)=√3,计算y=(a^√3)-a=10^√3-10=10^1.732-10=53.95-10=43.95.或y=a(a^(√3-1)-1)=10×(10^(1.732-1)-1)=10×(10^0.732-1)=10×(5.395-1)=10×4.395=43.95.
计算log(a)y=log(10)43.95=1.643.
计算答案：log(a)y=1+log(a)(a^(√3-1)-1)=1+log(10)(10^(√3-1)-1)=1+log(10)(10^(1.732-1)-1)=1+log(10)(10^0.732-1)=1+log(10)(5.395-1)=1+log(10)4.395=1+0.643=1.643.验证正确.