一道用换底公式解的题目在Rt△ABC中,c是斜边,a、b是直角边.求证:logc+b(a)+logc-b(a)=2logc+b(a)*logc-b(a)(注:c+b c-b均为底 "(a)"为真数 )
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![一道用换底公式解的题目在Rt△ABC中,c是斜边,a、b是直角边.求证:logc+b(a)+logc-b(a)=2logc+b(a)*logc-b(a)(注:c+b c-b均为底](/uploads/image/z/4884248-56-8.jpg?t=%E4%B8%80%E9%81%93%E7%94%A8%E6%8D%A2%E5%BA%95%E5%85%AC%E5%BC%8F%E8%A7%A3%E7%9A%84%E9%A2%98%E7%9B%AE%E5%9C%A8Rt%E2%96%B3ABC%E4%B8%AD%2Cc%E6%98%AF%E6%96%9C%E8%BE%B9%2Ca%E3%80%81b%E6%98%AF%E7%9B%B4%E8%A7%92%E8%BE%B9.%E6%B1%82%E8%AF%81%EF%BC%9Alogc%2Bb%28a%29%2Blogc-b%28a%29%3D2logc%2Bb%28a%29%2Alogc-b%28a%29%28%E6%B3%A8%EF%BC%9Ac%2Bb+c-b%E5%9D%87%E4%B8%BA%E5%BA%95+%22%28a%29%22%E4%B8%BA%E7%9C%9F%E6%95%B0+%EF%BC%89)
一道用换底公式解的题目在Rt△ABC中,c是斜边,a、b是直角边.求证:logc+b(a)+logc-b(a)=2logc+b(a)*logc-b(a)(注:c+b c-b均为底 "(a)"为真数 )
一道用换底公式解的题目
在Rt△ABC中,c是斜边,a、b是直角边.
求证:logc+b(a)+logc-b(a)=2logc+b(a)*logc-b(a)
(注:c+b c-b均为底 "(a)"为真数 )
一道用换底公式解的题目在Rt△ABC中,c是斜边,a、b是直角边.求证:logc+b(a)+logc-b(a)=2logc+b(a)*logc-b(a)(注:c+b c-b均为底 "(a)"为真数 )
c²=a²+b²
a²=(c+b)(c-b)
取对数
2lga=lg(c+b)+lg(c-b)
左边=lga/lg(c+b)+lga/lg(c-b)
通分=lga{[lg(c-b)+lg(c+b)]}/[lg(c-b)*lg(c-b)]
=lga*2lga/[lg(c-b)*lg(c-b)]
=2[lga/lg(c-b)]*[lga/lg(c-b)]
=2log(c-b)(a)*log(c+b)(a)
=右边
命题得证
换底:
左边=loga(a)/loga(c+b)+loga(a)/loga(c-b)
=(loga(c+b)+loga(c-b))/loga(c+b)loga(c-b)
=loga(c平方-b平方)/loga(c+b)loga(c-b)
=loga(a平方)/loga(c+b)loga(c-b)
=2/loga(c+b)loga(c-b)
右边=2*loga(a)/loga(c+b)*loga(a)/loga(c-b) = 2/loga(c+b)loga(c-b)
所以左边=右边
得证