How do you solve #(x-1)^[log(x-1)]=100(x-1)#?

1 Answer
Jun 18, 2016

#x = 1 + e^{1/2 (1 + sqrt[1 + 4 Log_e(100)])}#

Explanation:

Making #y = x-1# we have

#y^{log_e y} = 100 y->y^{log_e y-1} = 100#

Applying #log# to both sides

#(log_e y - 1)log_e y = log_e100#

Solving for #log_e y# we have

#log_e y = 1/2 (1 pm sqrt[1 + 4 Log_e(100)])#

so

#Y = x-1=e^{1/2 (1 pm sqrt[1 + 4 Log_e(100)])}#

Finally considering that #x -1> 0#

#x = 1 + e^{1/2 (1 + sqrt[1 + 4 Log_e(100)])}#