1. ## a^x = x

How can we algebraically find the intersection of an exponential function y=a^x with the straight line y=x? In other words, what is the solution to the equation a^x = x ?

2. Originally Posted by hakanaras
How can we algebraically find the intersection of an exponential function y=a^x with the straight line y=x? In other words, what is the solution to the equation a^x = x ?
First there is no guarantee that there are any real solutions to $a^x=x$. In fact if $\ln(a)>1$ it is obvious that there are no real solutions.

If $a<1$, then put $b=1/a$ and the equation becomes:

$e^{-\ln(b)x}=x$

which we can rearrange to:

$[\ln(b)~x]~e^{\ln(b)~x}=\ln(b)$

and this has solution:

$x=\frac{W(\ln(b))}{\ln(b)}$

or equivalently:

$x=-~\frac{W(-\ln(a))}{\ln(a)}$

where $W$ is Lambert's W function.

(If we are not concerned with real solutions then the above works period, though you have to keep an eye on the multiple values of the W function)

RonL

3. Originally Posted by CaptainBlack

:
:

or equivalently:

$x=-~\frac{W(-\ln(a))}{\ln(a)}$

where $W$ is Lambert's W function.

(If we are not concerned with real solutions then the above works period, though you have to keep an eye on the multiple values of the W function)

RonL
To demonstrate this working see this:

Code:
This is EULER, Version 2.3 RL-06.

Type help(Return) for help.
Enter command: (16777216 Bytes free.)

Processing configuration file.

Done.
>
>a=0.1;
>x=-LambertW(-log(a))/log(a)
0.399013
>
>
>
>a^x-x
5.55112e-017
>
RonL

4. Some replies do not deserve an unthanked fate. For my money, the above two are in that class. Hence the pre-emptive safe-guard.

(Would've added rep but apparently I've got to spread a bit more first).

And this thread is certainly as good as any to direct the interested reader to the following references:

http://www.cs.uwaterloo.ca/research/tr/1993/03/W.pdf

New analytic solution to classic problem?

5. Originally Posted by mr fantastic
Some replies do not deserve an unthanked fate. For my money, the above two are in that class. Hence the pre-emptive safe-guard.

(Would've added rep but apparently I've got to spread a bit more first).

And this thread is certainly as good as any to direct the interested reader to the following references:

http://www.cs.uwaterloo.ca/research/tr/1993/03/W.pdf

New analytic solution to classic problem?
Now you have embarrassed me

Great references.

RonL