razorfane Posted March 8, 2005 Share Posted March 8, 2005 Hi, i'm new, and i wanted to share a very simple algebra problem i like. [math]x^{2x}=2x[/math] It's very simple, but not as easy as it looks, if you know the answer right away, give others a chance. Link to comment Share on other sites More sharing options...
Primarygun Posted March 8, 2005 Share Posted March 8, 2005 Hihi. To find an exact value, is calculus knowledge required? Link to comment Share on other sites More sharing options...
jcarlson Posted March 8, 2005 Share Posted March 8, 2005 wheres the problem? lol edit: nevermind... it showed up after i replied... wierd. Link to comment Share on other sites More sharing options...
The Rebel Posted March 8, 2005 Share Posted March 8, 2005 The answer is so simple yet I can't figure out a way of calculating it algebraicly, unless I'm missing something stupid. Link to comment Share on other sites More sharing options...
razorfane Posted March 8, 2005 Author Share Posted March 8, 2005 No, it's not calculus, it's algebra and it is pretty simple, but not easy. You probably realized it already but you can't isolate the x, so it's not done by regular ecuation solving. Think. Link to comment Share on other sites More sharing options...
ydoaPs Posted March 8, 2005 Share Posted March 8, 2005 i got .25 and 1.4361702. Link to comment Share on other sites More sharing options...
razorfane Posted March 9, 2005 Author Share Posted March 9, 2005 i got .26595745 and 1.4361702.i'm not sure about that' date=' but the gradient is [math']y=x^{2x}-2x\rightarrow\frac{dy}{dx}=2x^{2x+1}-2[/math]. You're complicating yourself, it's 100% algebra. Link to comment Share on other sites More sharing options...
CPL.Luke Posted March 9, 2005 Share Posted March 9, 2005 I got so far as x^2x-1=2 I got there using logarithms edit just got to x 1n(x)= 1n (3)/-2 edit slight error in math 1n(x)= 1n(3)/2x-1 Link to comment Share on other sites More sharing options...
Ducky Havok Posted March 9, 2005 Share Posted March 9, 2005 I know one answer is 1/4.... the other is approximately 1.4437374... I have no clue how to solve it algebraically though, good problem! Link to comment Share on other sites More sharing options...
CPL.Luke Posted March 9, 2005 Share Posted March 9, 2005 where did you get those answers from? Link to comment Share on other sites More sharing options...
ydoaPs Posted March 9, 2005 Share Posted March 9, 2005 i got my answers by graphing it. Link to comment Share on other sites More sharing options...
razorfane Posted March 10, 2005 Author Share Posted March 10, 2005 I know one answer is 1/4.... umm, yeah, the answer is 1/4, but off course the beauty lies in solving it algebraically. Link to comment Share on other sites More sharing options...
matt grime Posted March 11, 2005 Share Posted March 11, 2005 so what is the algebraic method then? Link to comment Share on other sites More sharing options...
Zone Ranger Posted March 11, 2005 Share Posted March 11, 2005 i got .26595745 and 1.4361702.i'm not sure about that' date=' but the gradient is [math']y=x^{2x}-2x\rightarrow\frac{dy}{dx}=2x^{2x+1}-2[/math]. I just had to point out that the above is not correct. The derivative of [math]x^{2x}[/math] is [math]2(ln(x)+1)x^{2x}[/math] Link to comment Share on other sites More sharing options...
razorfane Posted March 11, 2005 Author Share Posted March 11, 2005 so what is the algebraic method then?That's what I'm asking you guys. It took me and 2 other dudes like 3 weeks to solve it, and this is the simplest problem of it's type. Link to comment Share on other sites More sharing options...
ydoaPs Posted March 11, 2005 Share Posted March 11, 2005 i solved it graphically. i used my calculator this time and got closer values than before. they are .25 and aproxamately 1.443737. i set it =0 and assumed y=0. my answers are where the function crosses the x-axis. Link to comment Share on other sites More sharing options...
Ducky Havok Posted March 11, 2005 Share Posted March 11, 2005 that's how I did it for my answers... I gave it to my highschool calculus teacher and she can't figure it out either... I haven't given up though! I'm still trying Link to comment Share on other sites More sharing options...
ydoaPs Posted March 11, 2005 Share Posted March 11, 2005 so razor, how did u do it? Link to comment Share on other sites More sharing options...
razorfane Posted March 11, 2005 Author Share Posted March 11, 2005 so razor, how did u do it?You're giving up that easy? Really, it's just a matter of imagination. Link to comment Share on other sites More sharing options...
ydoaPs Posted March 11, 2005 Share Posted March 11, 2005 i already got the anwers, i want to know your simple yet hard algebraic method. Link to comment Share on other sites More sharing options...
razorfane Posted March 11, 2005 Author Share Posted March 11, 2005 I know you got the answer. are you insinuating there's no other way? Link to comment Share on other sites More sharing options...
ydoaPs Posted March 11, 2005 Share Posted March 11, 2005 no. did you not read my post? i just want to know your way. Link to comment Share on other sites More sharing options...
razorfane Posted March 11, 2005 Author Share Posted March 11, 2005 Oh, ok. Actually there is no algebraic solution and... i'm joking, here it is: [math]x^{2x}=2x[/math] [math]x^{2x-1}=2[/math] [math]x^{\frac{2x-1}{2}}=2^{\frac{1}{2}}[/math] [math]x^{x-\frac{1}{2}}=2^{\frac{1}{2}}[/math] [math]x^{x-\frac{1}{2}}=\left(\tfrac{1}{2}\right)^{-\frac{1}{2}}[/math] [math]x^{x-\frac{1}{2}}=\left(\tfrac{1}{4}\right)^{-\frac{1}{4}}[/math] [math]x^{x-\frac{1}{2}}=\left(\tfrac{1}{4}\right)^{\frac{1}{4}-\frac{1}{2}}[/math] Since both members are structurally the same, you deduct: [math]X=\frac{1}{4}[/math] Link to comment Share on other sites More sharing options...
Ducky Havok Posted March 12, 2005 Share Posted March 12, 2005 okay, I understand everything up until this point. [math]x^{x-\frac{1}{2}}=\left(\tfrac{1}{4}\right)^{\frac{1}{4}-\frac{1}{2}}[/math] Will you please explain how you got from the previous step to that one? Also, what about the other answer, 1.4437374? Is there a way to get that one? Link to comment Share on other sites More sharing options...
Ducky Havok Posted March 12, 2005 Share Posted March 12, 2005 nevermind, I didn't see the negative in the previous one and that completely messed me up until I did it on paper. I get it now, -1/4 is equal to 1/4-1/2. I'm still curious about the other answer (1.4437374) though Link to comment Share on other sites More sharing options...
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