Logarithm Equation

Date: 01/22/2002 at 14:28:30
From: Timmy
Subject: A strange logarithm equation

This equation, which I've tried to solve, has turned out to be 
impossible for my skills!

   2x^2-1 = ln(2x^2) 

If there is some way to solve this I'd be really grateful to know.

Date: 01/22/2002 at 14:40:07
From: Doctor Paul
Subject: Re: A strange logarithm equation

After fiddling with this for a minute, it occured to me that this was 
a pretty hard problem unless it was designed to be solved. And as it 
turns out, this problem is designed to be solved. Here's how to do it:

I figured I had nothing to lose if I just decided to see where the 
left-hand side and the right-hand side had roots. If they both have 
roots at the same place, then I've found a solution to the problem.

   2x^2 - 1 = 0

   x = +- 1/sqrt(2) = +- sqrt(2)/2


ln(2x^2) = 0 is true when 2x^2 = 1 and that implies

   x = +- 1/sqrt(2) = +- sqrt(2)/2

So it turns out that the two solutions to this problem are 

   x = +- sqrt(2)/2.

Sometimes it helps to know that these sorts of problems will have 
solutions. So finding them might involve unusual techniques. But if 
you know there is a solution and it's going to be a reasonable 
solution, then finding it usually isn't too difficult.

I hope this helps.  Please write back if you'd like to talk about this 
more.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/