March 1st, 2012, 03:23 AM  #1 
Newbie Joined: Feb 2012 Posts: 18 Thanks: 0  Stumped  Exponential Equation
Here is the problem. Find all values of such that . I can see by inspection that the solution is . But I cannot see a way to solve this algebraically. Does anybody know what method I need to solve this?

March 1st, 2012, 04:08 AM  #2 
Senior Member Joined: Aug 2011 Posts: 334 Thanks: 8  Re: Stumped  Exponential Equation
The study of the function y(x)=x+(3^x)4 leads to : first : the function is strictly increassing. second : there is only one root y(1)=0. Then, there is no difficulty to answer to the question. 
March 1st, 2012, 04:12 AM  #3 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  Re: Stumped  Exponential Equation [color=#000000]Let (continuous in ) it is obvious that . Studying the monotony of f, by computing the first derivative it is obvious that f is strictly increasing in and thus 11, which means that f gets the value 4 for only x=1. So f(x)<4 for x<1.[/color] 
March 1st, 2012, 05:28 AM  #4 
Newbie Joined: Feb 2012 Posts: 18 Thanks: 0  Re: Stumped  Exponential Equation
Thanks for these responses. I used the same idea of monotonicity to decide that the solution was . I just wondered if there was any algebraic method.


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