I have an equation that I cannot solve, can anyone time how to get to $x=0.1225$?

Question

How do I solve the following equation?: $$ 0.2 = 0.819 \exp(-11.5x)+0.0975 \exp(-70.1x)+0.0325 \exp(-179x) $$ The result is supposed to be $x \approx 0.1225$

Answer 1

The huge negative exponents will make the last two terms on the right very small, so you can get a preliminary value by ignoring them. We get $\ln \frac {0.2}{0.819}\approx -11.5x, x \approx 0.122588$ Now you can incorporate the other terms in a fixed point iteration $$0.2 = 0.819 \exp(-11.5x)+0.0975 \exp(-70.1x)+0.0325 \exp(-179x)\\ \frac {-1}{11.5}\log \left(\frac 1{0.819}\left(0.2-0.0975 \exp(-70.1x_i)0.0325 \exp(-179x_i)\right)\right)=x_{i+1}$$ The iteration will converge very rapidly