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对数里有下面这祥一个性质: “若对数式log_ab=c恒成立,一般地有log_(a~n)~(b~n)=c,这里的n∈R,且n≠0”。 [证明] log_ab=c(?)b=a~c■b~n=(a~n)~c 在n≠0时,两边同取以a~n为底的对数, 则有: log_(a~n)~(b~n)=c,n∈R且n≠0 运用上述性质,可解决一些较为复架的对数问题,现举几例如下。 [例1] 已知log_8(x~2+1)~3-log_2xy+log_(2~(1/2))·(y~2+4)/~(1/2)=3 试确定x,y之值 (85年常州初中数学竞赛题) 分析:初中数学竞赛一般不要求换底公式,上述问题即使用换底公式,也颇费周折,若联想到上述性质,则解法较为简捷。
There are the following properties in the logarithm: “If the logarithm log_ab=c is constant, generally log_(a~n)~(b~n)=c, where n∈R, and n≠0”. [Prove] log_ab=c(?)b=a~c■b~n=(a~n)~c When n≠0, both sides take the logarithm of a~n as the base, then there are: log_( a~n)~(b~n)=c,n∈R and n≠0 Using the above properties, we can solve the logarithmic problems of some complex frames. [Example 1] It is known that log_8(x~2+1)~3-log_2xy+log_(2~(1/2))·(y~2+4)/~(1/2)=3 determines x, The value of y (85 years of the Changzhou junior high school math contest title) Analysis: The junior high school math competition generally does not require a change in the bottom formula. The above problem is the use of the formula for changing the base, and it is also a complicated process. If you think of the above properties, the solution is simpler.