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Solve e^x=x^e [Solved!]

My question

How to solve e^x=x^e?

Relevant page

Exponential & Logarithmic Functions

What I've done so far

It could not be solved by the quadratic equation.

X

How to solve e^x=x^e?
Relevant page

<a href="/exponential-logarithmic-functions/exponential-log-functions-intro.php">Exponential &amp; Logarithmic Functions</a>

What I've done so far

It could not be solved by the quadratic equation.

Re: Solve e^x=x^e

Hi James

Well, your problem cannot be solved using quadratic equations (because it isn't a quadratic equation). Actually, it can't be solved using ordinary algebra at all.

My first thought was to take log of both sides, but that doesn't give us anything we can solve:

`"LHS" = ln e^x = x ln e = x`

`"RHS" = e ln x`

So I would suggest using graphs, like I do on this page:

5. Graphical Solution of non-linear Systems

Your "system" is actually

`y = e^x`
`y = x^e`

Another way to do it would be to graph

`y = e^x - x^e` and see where it intersects the x axis.

Hope that helps.

X

Hi James

Well, your problem cannot be solved using quadratic equations (because it isn't a quadratic equation). Actually, it can't be solved using ordinary algebra at all.

My first thought was to take log of both sides, but that doesn't give us anything we can solve:

`"LHS" = ln e^x = x ln e = x`

`"RHS" = e ln x`

So I would suggest using graphs, like I do on this page:

<a href="/systems-of-equations/5-graphical-solution-non-linear-system.php">5. Graphical Solution of non-linear Systems</a>

Your "system" is actually

`y = e^x`
`y = x^e`

Another way to do it would be to graph

`y = e^x - x^e` and see where it intersects the x axis.

Hope that helps.

Re: Solve e^x=x^e

ok, I'll try the graph thing here.

First method (graphing `y=e^x` and `y=x^e` on the same axes):

Forum graph - svgphp-01234-151015202530xy

It's not easy to see the solution there.

Second method (graphing `y=e^x - x^e`:

Forum graph - svgphp-111.11.21.31.41.51.61.71.81.922.12.22.32.42.52.62.72.82.90.10.20.30.40.50.60.70.80.911.11.21.31.41.51.61.71.81.9-0.1-0.2-0.3-0.4-0.5x

It looks like the answer is around `x=2.7`

Checking: `e^2.7 ~~ 14.8797`

`2.7^e = 14.8788`

Close enough!

Thanks a lot.

X

ok, I'll try the graph thing here.

First method (graphing `y=e^x` and `y=x^e` on the same axes):

[graph]310,250;-1,4;-1,30,1,5;e^x,x^e[/graph]

It's not easy to see the solution there.

Second method (graphing `y=e^x - x^e`:

[graph]310,250;1,3;-0.5,2,0.1,0.1;e^x-x^e[/graph]

It looks like the answer is around `x=2.7`

Checking: `e^2.7 ~~ 14.8797`

`2.7^e = 14.8788`

Close enough!

Thanks a lot.

Re: Solve e^x=x^e

Well done, James! You're very welcome.

X

Well done, James! You're very welcome.

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