关于方程a^x=log_ax解的讨论

Discussion About solution of Equation a^x= log_ax

本文采用将两条曲线y =ax 与y =logax的交点坐标和底数分别表示成参数k(k是直线斜率 )的函数的方法 ,系统讨论了方程ax=logax有解的各种充分条件 .对于改进高等数学教学具有一定的现实意义 .

How many solutions are there for the equation a x= log ax ?By means of the slope k of the straight line y=kx,we obtain repress entation of x(k),y(k) and a(k),then it turns out that:(i) for 0<a<e -e the equation has exactly three solutions;(ii) for e -e <a<1 there is only one solution,and (iii) for a=e -e ,the two curves are tangent to one another,so for this value of x there is also a single root on the line y=x.