Finding x Without a Graph: e^x = x^10

[JasonRox]

The text wants me to find x using a graph, but since I don't like taking my sweet time building one so accurate to find an answer to one decimal place, I rather find x without the graph.

This is it:

e^x = x^10

This isn't important or anything, but I figured that I can use some practice. I was thinking of using logs, and then switch the bases so that they are all the same (e or 10).

Is this the way to go?

Don't jump out with an answer or that blows everything. I would like to give it a shot.

I'll be gone to school for the next 12-14 hours, so I'll probably have it by tonight. I might also hop on a computer at school if they aren't so busy.

Thanks for any help.

Note: I know graphs are important, and I build rough ones all the time. I normally visualize graphs using simple translation rules, and odd/even functions. Building one so details to find x to one decimal place is ridiculous.

Note: If you want to blow it and jump out with an answer, don't bother because I can just look at the back of the text for that.

 

[matt grime]

At the risk of having someone point out something I'm not aware of, there is no analytic way to solve that equation: logs ain't going to help.

 

[DeadWolfe]

Matt is correct, which is why they want you to solve this with a graph.

No amount of algebra or calculus can help you.

Do you have a graphing calculator, that'll solve it for you.

 

[robphy]

You could try to do it numerically, of course.

Since you want to "find an answer to one decimal place", I'm sure there is an approximation [valid in the neighborhood of the answer] that one could do.

 

[Zurtex]

You can take logs of both sides and make an iterative formulae, but like said before you can't get an 'exact' answer as such.

 

[JasonRox]

Wow!

Now I don't feel like an idiot anymore.

There has to be a way though.

 

[Casio]

Use NewtonRaph

 

[mathwonk]

i don't get it. if e^x = x^10 then since x^10 = e^10ln(x), we have x = 10ln(x), so x/ln(x) = 10. oh isee, I'm in trouble now.

well the answer is obviously negative, so assume instead that x^10 = 1/e^x, so when x= 0 the rhs side is bigger but when x = 1, the lhs is bigger, so the answer is between 0 and 1, (i.e. the original answer is between 0 and -1). it should not take forever to get it to within 1/10 this way without graphing it.

 

[JasonRox]

I know you can do it, but I like concrete answers. :)

 

[mathwonk]

i just showed you how to get one. i.e. i just gave you the method ("intermeduiate value theorem") and used it to get the integers digit for you. you said you did not want us to give the whole problem away. do you need more?