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September 5th, 2014, 01:02 PM  #1 
Senior Member Joined: Nov 2013 Posts: 434 Thanks: 8  solve exponential inequality
solve 3^(x+1/x)<=4

September 5th, 2014, 02:13 PM  #2 
Newbie Joined: Sep 2014 From: us Posts: 10 Thanks: 1 
Step 1: Take the common logarithm or natural logarithm of each side. Step 2: Use the properties of logarithms to rewrite the problem. Specifically, use Property which says log(x^y) = y log x Step 3: Divide each side by the logarithm. Step 4: Use a calculator to find the decimal approximation of the logarithms. Step 5: Finish solving the problem by isolating the variable... 
September 5th, 2014, 02:36 PM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,674 Thanks: 2654 Math Focus: Mainly analysis and algebra 
I don't agree with step four. In step 5 you must take care to note whether $x$ is greater or less than zero. 
September 5th, 2014, 02:54 PM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,958 Thanks: 1146 Math Focus: Elementary mathematics and beyond 
$\displaystyle 3^{x+\frac1x}\le4$ so we must have $\displaystyle \frac{x^2+1}{x}\le\log_34$ Since $\displaystyle \frac{x^2+1}{x}$ may be written as $\displaystyle \frac{(x1)^2}{x}+2$ the minimum of $\displaystyle \frac{x^2+1}{x}$ is 2, for $\displaystyle x>0$. Hence the original statement is only true for x < 0. 
September 5th, 2014, 03:13 PM  #5 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  

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