解方程6*25^x=9^x+15^x

题目

解方程6*25^x=9^x+15^x

答案

6*25^x = 9^x + 15^x
6*5^(2x) = 3^(2x) + 3^x*5^x
6*5^x/3^x = 3^x/5^x + 1,上下除以3^x*5^x,∵15^x≠0
6*(5/3)^x = (3/5)^x + 1
6 = (3/5)^x/(5/3)^x + (3/5)^x
(3/5)^(2x) + (3/5)^x - 6 = 0
令u=(3/5)^x
u² + u - 6 = 0
(u-2)(u+3) = 0
u = 2 or u = -3
(3/5)^x = 2 or (3/5)^x = -3,(舍)
x = log2/log(3/5)
=log2 / (log3 - log5)