点解log x^k= k log x
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发布时间：2023-09-01 00:59
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时间：2024-11-12 14:03
log x^(1) = (1) log x
it is true for n=1
Assume it is true for n= k
where k belongs to nature number
i.e. log x^k = k log x Consider n=k+1
log x^(k+1) = log x^k x = log x^k + log x = (k+1)logx it is true for n=k+1
By Mathematical inction
it is true for all positive integer n. For n=0
log x^0 = log 1 = 0 = (0) log x it is ture all positive integers. For negetivei integer
let n=-m
log x^n = log x^-m = log 1/x^m = log 1- log x^m =- log x^m = -m log x it is true for all integers. 2009-07-31 23:20:46 补充： For rational number
let n=a/b
where a and b are integer and no mon factor . by the above result
we know that log x^a=a log x and log x^b = blogx
2009-07-31 23:23:40 补充： Consider blogx^n = log x^(nb) = log x^a = alogx i.e. log x^n = (a/b)logx = nlogx so it is true for all rational numbers .
log(-2)^4=4log(-2)
定理 : log a*b = log a+ log b log a*b*c = log a+ log b+ log c log a*a =" log a^2 "= log a+ log a=" 2 log a " log x^k = log x*x*x*x*...x (共乘k次) = log x+ log x+ ... log x (加k次) = k log x
你知道lo *** 解吗？假设： logX Y = Z； XZ = Y； X的Z次方等于Y，这是log的简单解释。 假如： XZ = Y； 双边都以power of k来剩，即： XZ(k)= Yk； 亦即是： k (logX Y)； 或 k(Z)
参考: 个人意见
