高一数学题：解方程6^x＋4^x＝9^x 解方程6^x+4^x=9^x
\u9ad8\u4e00\u6570\u5b66\u9898\uff1a\u89e3\u65b9\u7a0b6^x\uff0b4^x\uff1d9^x\uff0c9^\uff08-x\uff09-2*3^\uff081-x\uff09=276^x\uff0b4^x\uff1d9^x
3^x*2^x+(2^x)^2=(3^x)^2,
\u4e24\u8fb9\u540c\u9664\u4ee5(2^x)^2\uff0c
\uff083/2\uff09^x+1=[(3/2)^x]^2,
\u8bbe\uff083/2\uff09^x=t,
t+1=t^2,
t^2-t-1=0,
t=(1+\u221a5\uff09/2\uff0c\uff08\u820d\u53bb\u8d1f\u6839\uff09\uff0c
\uff083/2\uff09^2=(1+\u221a5\uff09/2,\u4e24\u8fb9\u53d6\u5bf9\u6570\uff0c
x=[lg(1+\u221a5\uff09-lg2]/(lg3-lg2).
9^\uff08-x\uff09-2*3^\uff081-x\uff09=27
1/[\uff083^2)^x]-6/(3^x)=27,
\u8bbe1/3^x=t,
t^2-6t-27=0
(t-9)(t+3)=0
t=9,\u6216t=-3(\u4e0d\u7b26\u5408\u9898\u610f\uff0c\u820d\u53bb\uff09
1/3^x=9,
3^(-x)=3^2
-x=2
x=-2
\u7ecf\u68c0\u9a8cx=-2\u662f\u539f\u65b9\u7a0b\u7684\u89e3\u3002
\u3010\u89e3\u9898\u601d\u8def\u3011
6^x=(2*3)^x
4^x=(2*2)^x
9^x=(3*3)^x
\u540c\u9664\u4ee5(2*3)^x,\u52194^x\u53d8\u6362\u4e3a(2/3)^x, 9^x\u53d8\u6362\u4e3a\uff083/2)^x=((2/3)^x)^(-1)
\u3010\u89e3\u7b54\u3011
\u65b9\u7a0b\u4e24\u8fb9\u540c\u9664\u4ee56^x
1+(2/3)^x=((2/3)^x)^(-1)
\u8bbet=(2/3)^x (\u6839\u636e\u5e42\u6307\u6570\u51fd\u6570\u6027\u8d28\uff0c\u6709t>0)
\u5219\u539f\u65b9\u7a0b\u53d8\u4e3a
1+t=1/t
t²+t-1=0
t=\uff08\u6839\u53f75-1\uff09/2 (\u7531t>0,\u5254\u9664\u8d1f\u6839\uff09
\u5219(2/3)^x=\uff08\u6839\u53f75-1\uff09/2
\u3010\u53d6\u5bf9\u6570\uff0c\u4ee52/3\u4e3a\u5e95\u3011
x=log[(2/3),\uff08\u6839\u53f75-1\uff09/2] (\u524d\u8005\u4e3a\u5e95\uff0c\u540e\u8005\u4e3a\u771f\u6570)
\u3010\u7b54\u6848\u3011\u3010x=log[(2/3),\uff08\u6839\u53f75-1\uff09/2] \u3011
3^x*2^x+(2^x)^2=(3^x)^2,
两边同除以(2^x)^2，
（3/2）^x+1=[(3/2)^x]^2,
设（3/2）^x=t,
t+1=t^2,
t^2-t-1=0,
t=(1+√5）/2，（舍去负根），
（3/2）^2=(1+√5）/2,两边取对数，
x=[lg(1+√5）-lg2]/(lg3-lg2).
解:
∵恒有: 9^x≠0
∴原方程两边同除以9^x.整理可得:
[(2/3)^x]+[(2/3)^x]²=1
可设k=(2/3)^x.
则k＞0且k²+k-1=0
解得k=(-1+√5)/2
∴(2/3)^x=(-1+√5)/2
两边取对数,可得:
xln(2/3)=ln[(-1+√5)/2]
∴x={ln[(-1+√5)/2]}/[ln(2/3)]
解方程6^x＋4^x＝9^x
解：方程两边同除以4^x得(3/2)^x+1=(3/2)^(2x)
即有[(3/2)^x]²-(3/2)^x-1=0
令(3/2)^x=u，则有u²-u-1=0
故u=(3/2)^x=(1+√5)/2 (负根舍去)
∴x=log‹3/2›[(1+√5)/2]
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