My Math Forum Stumped - Exponential Equation

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 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 $x$ such that $x+3^{x}<4$. I can see by inspection that the solution is $x < 1$. 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 $f(x)=3^x+x$ (continuous in $\mathbb{R}$) it is obvious that $f(1)=3^1+1=4$. Studying the monotony of f, by computing the first derivative $f'(x)=1+\ln(3)\cdot 3^x>0, \forall x \in \mathbb{R}$ it is obvious that f is strictly increasing in $\mathbb{R}$ and thus 1-1, 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 $x<1$. I just wondered if there was any algebraic method.

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