SOLUTION: x^logx+1 =10^6

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Question 958155: x^logx+1 =10^6
Found 2 solutions by Fombitz, Edwin McCravy:
Answer by Fombitz(32388) About Me  (Show Source):
Answer by Edwin McCravy(20065) About Me  (Show Source):
You can put this solution on YOUR website!
The other tutor interpreted your log(x)+1 as log(x+1) making it impossible
to solve by algebra methods, but only by graphing calculator -- or iterative
methods, which is what the graphing calculator is programmed to do.  

Interpreting it as I think you intended:

x%5E%28log%28%28x%29%29%2B1%29%22%22=%22%2210%5E6
 
Take logs (base 10) of both sides
 
log%28%28x%5E%28log%28%28x%29%29%2B1%29%29%29%22%22=%22%22log%28%2810%5E6%29%29
 
Simplify by using the rule of logs: log%28%28A%5EB%29%29=B%2Alog%28A%29
 
%28log%28%28x%29%29%2B1%29log%28%28x%29%29%22%22=%22%226log%28%2810%29%29
 
Distribute the left side.
Simplify the right side using the rule log%28%2810%29%29=1.
 
log%28%28x%29%29log%28%28x%29%29%2Blog%28%28x%29%29%22%22=%22%226%2A1
 
%28log%28%28x%29%29%5E%22%22%29%5E2%2Blog%28%28x%29%29%22%22=%22%226
 
%28log%28%28x%29%29%5E%22%22%29%5E2%2Blog%28%28x%29%29-6%22%22=%22%22%220%22

Let u = log(x)

u%5E2%2Bu-6%22%22=%22%22%220%22

%28u%2B3%29%28u-2%29%22%22=%22%22%220%22

  u+3 = 0;      u-2 = 0
    u = -3        u = 2
log(x) = -3;   log(x) = 2
  x = 10-3        x = 102
  x = 1%2F10%5E3        x = 100 
  x = 1%2F1000

Two solutions, 1%2F1000 and 100

Edwin