Can someone help me solve this
$$e^{-x}=\ln(1/x)$$
I think these two functions are inverses of each other and since they both meet the line $y=x$ they intersect
But I'm not able to simplify the equality above to find their point of intersection
Help me, please.
On
There is no analytical solution to this equation even using special functions. Then you will need some numerical method for finding the zero of function $$f(x)=e^{-x}+\log(x)$$ Its derivative $$f'(x)=-e^{-x}+\frac 1x$$ cancels when $x=-W(-1)$ which is a complex number; so, $f'(x)>0 $and there is only one root.
By inspection, $f(1)=\frac 1e >0$. So, using Newton method with $x_0=1$ will give the following iterates $$\left( \begin{array}{cc} n & x_n \\ 0 & 1.0000000000 \\ 1 & 0.4180232931 \\ 2 & 0.5413727558 \\ 3 & 0.5664266698 \\ 4 & 0.5671427445 \\ 5 & 0.5671432904 \end{array} \right)$$
We could have generated a better estimate of the starting point building first the simplest Padé approximant at $x=1$ $$f(x) \simeq \frac{\frac{1}{e}+\frac{(2 e-1) }{2 e}(x-1)}{1+\frac{1}{2}(x-1)}$$ giving $x_0=\frac{2 e-3}{2 e-1}\approx 0.549201$.
Continuing with $[1,n]$ Padé approximants, it will be closer and closer to the solution as shown below $$\left( \begin{array}{ccc} 1 & \frac{-3+2 e}{-1+2 e} & 0.5492006529 \\ 2 & \frac{-8+34 e-36 e^2+12 e^3}{-2+16 e-24 e^2+12 e^3} & 0.5651368901 \\ 3 & \frac{30-300 e+636 e^2-504 e^3+144 e^4}{6-108 e+348 e^2-360 e^3+144 e^4} & 0.5670287743 \\ 4 & \frac{-144+3216 e-12480 e^2+18000 e^3-11520 e^4+2880 e^5}{-24+1056 e-5520 e^2+10800 e^3-8640 e^4+2880 e^5} & 0.5671134681 \end{array} \right)$$
On
As you observed, the functions $f(x)=e^{-x}$ and $g(x)=\log(1/x)$ are inverses. If $f(x)$ intersects $y=x$ at $a$ then we have $f(a)=a$. Applying $g$ we get $a=g(a)$. Then $f(a)=g(a)$ and $a$ is a solution for the equation. So we have $a = f(a) = e^{-a}$, which is equivalent to $a e^a = 1$. This is precisely the equation that is solved by the Lambert $W$ function. Apply $W$ to get $a = W(1)$
Alpha gives a numeric solution $x \approx 0.56714$ I don't think it surprising that it did not give an analytic result. This looks like an equation that will not yield to algebra.