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March 1st, 2012, 03:23 AM   #1
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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?
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March 1st, 2012, 04:08 AM   #2
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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.
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March 1st, 2012, 04:12 AM   #3
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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 1-1, which means that f gets the value 4 for only x=1. So f(x)<4 for x<1.[/color]
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March 1st, 2012, 05:28 AM   #4
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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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