求关于 lnα的常用计算公式

[求关于除夕的古诗]除夜【宋】文天祥乾坤空落落，岁月去堂堂；末路惊风雨，穷边饱雪霜。命随年欲尽，身与世俱忘；无复屠苏梦，挑灯夜未央。元日【宋】王安石爆竹声中一岁除，春风送暖如屠苏。千门万...+阅读

对数的性质及推导用^表示乘方，用log(a)(b)表示以a为底，b的对数*表示乘号，/表示除号定义式：若a^n=b(a>0且a≠1)则n=log(a)(b)基本性质：1.a^(log(a)(b))=b2.log(a)(MN)=log(a)(M)+log(a)(N);3.log(a)(M/N)=log(a)(M)-log(a)(N);4.log(a)(M^n)=nlog(a)(M)推导1.这个就不用推了吧，直接由定义式可得（把定义式中的[n=log(a)(b)]带入a^n=b)2.MN=M*N由基本性质1（换掉M和N）a^[log(a)(MN)]=a^[log(a)(M)]*a^[log(a)(N)]由指数的性质a^[log(a)(MN)]=a^{[log(a)(M)]+[log(a)(N)]}又因为指数函数是单调函数，所以log(a)(MN)=log(a)(M)+log(a)(N)3.与2类似处理MN=M/N由基本性质1（换掉M和N）a^[log(a)(M/N)]=a^[log(a)(M)]/a^[log(a)(N)]由指数的性质a^[log(a)(M/N)]=a^{[log(a)(M)]-[log(a)(N)]}又因为指数函数是单调函数，所以log(a)(M/N)=log(a)(M)-log(a)(N)4.与2类似处理M^n=M^n由基本性质1（换掉M）a^[log(a)(M^n)]={a^[log(a)(M)]}^n由指数的性质a^[log(a)(M^n)]=a^{[log(a)(M)]*n}又因为指数函数是单调函数，所以log(a)(M^n)=nlog(a)(M)其他性质：性质一：换底公式log(a)(N)=log(b)(N)/log(b)(a)推导如下N=a^[log(a)(N)]a=b^[log(b)(a)]综合两式可得N={b^[log(b)(a)]}^[log(a)(N)]=b^{[log(a)(N)]*[log(b)(a)]}又因为N=b^[log(b)(N)]所以b^[log(b)(N)]=b^{[log(a)(N)]*[log(b)(a)]}所以log(b)(N)=[log(a)(N)]*[log(b)(a)]{这步不明白或有疑问看上面的}所以log(a)(N)=log(b)(N)/log(b)(a)性质二：（不知道什么名字）log(a^n)(b^m)=m/n*[log(a)(b)]推导如下由换底公式[lnx是log(e)(x),e称作自然对数的底]log(a^n)(b^m)=ln(a^n)/ln(b^n)由基本性质4可得log(a^n)(b^m)=[n*ln(a)]/[m*ln(b)]=(m/n)*{[ln(a)]/[ln(b)]}再由换底公式log(a^n)(b^m)=m/n*[log(a)(b)]--------------------------------------------（性质及推导完）公式三：log(a)(b)=1/log(b)(a)证明如下：由换底公式log(a)(b)=log(b)(b)/log(b)(a)----取以b为底的对数，log(b)(b)=1=1/log(b)(a)还可变形得：log(a)(b)*log(b)(a)=1