I have a problem solving the below equation with respect to $x$:
$0.6\cdot \exp(\frac{-40}{x})+0.4 \cdot \exp(\frac{10}{x})=1$
My problem is that I have two exponential functions which are added together, so I don't know if I should use the natural log in order to get an expression solved with respect to $x$, or what I should do to accomplish this. I hope someone can help me with solving the expression above, so that I get an expression for $x$
David
Equations which are sums of exponential functions do not show analytical solutions in general.
Your case seems among the particular ones since writing $y=e^{\frac{10} x}$ the euation writes $$\frac{0.6}{y^4}+0.4y=1$$ which, unfortunately reduces to a quintic polynomial which is not the most pleasant to solve (few quintic equations have analytical solutions - if they have, they tend to show many radicals).
Probably the easiest could be to look for the zero of $$f(y)=0.4y^5-y^4+0.6$$ If you graph the function, you could notice that the positive root is around $2.5$. So, using Newton method for example $$f'(y)=2y^4-4y^3$$ gives the iterative scheme $$y_{n+1}=\frac{8 y_n^5-15 y_n^4-3}{10 (y_n-2) y_n^3}$$ Using $y_0=2.5$, the interates will then be $$y_1=2.46160000000000$$ $$y_2=2.45898401374040$$ $$y_3=2.45897234660929$$ $$y_4=2.45897234637796$$ which is the solution for fifteen significant figures.
Now, from $y$ get $x$.
Edit
The solution of the equation can be expressed with radicals. Be ready for the monster $$y=\frac{3}{8}+\frac{1}{8} \sqrt{25-\frac{16\ 5^{2/3}}{\sqrt[3]{\sqrt{65}-5}}+8 \sqrt[3]{5 \left(\sqrt{65}-5\right)}}+\frac{1}{2} \sqrt{\frac{25}{8}+\frac{5^{2/3}}{\sqrt[3]{\sqrt{65}-5}}-\frac{1}{2} \sqrt[3]{5 \left(\sqrt{65}-5\right)}+\frac{195}{8 \sqrt{25-\frac{16\ 5^{2/3}}{\sqrt[3]{\sqrt{65}-5}}+8 \sqrt[3]{5 \left(\sqrt{65}-5\right)}}}}$$