TITLE: Make x the subject of a double exponential equation
QUESTION [1 upvotes]: Many statistical packages allow a double exponential function to be fit to your data (below), yielding five constants A, B, C, R and S, which can be used to describe the curve.
$$ y = A + B * R^x + C * S^x $$
These types of curves are particularly useful to biologists, chemists and other non-mathematicians. Rearranging this equation to make x the subject would be particularly useful to estimate the x value needed to reach a given y value. However, rearranging this formula is beyond my maths skills. I imagine the solution would also be useful for others, who have searched but not found anything.

REPLY [0 votes]: Except for very few cases (such as $R=S^2$, $S^3$ or $S^4$ or the reciprocals which would lead to polynomials of degree $2$, $3$ ot $4$ in $R$ or $S$), there is no analytical solution to equations  $$y = A + B * R^x + C * S^x$$ and only numerical methods will provide $x$ from known values of $y,A,B,C,R,S$. Newton or some variants (Halley, Householder, ...) would be quite suitable but they require a "reasonable" starting value.
For illustration purposes, just consider the equation $$y=7+6\times 5^x +4\times 3^x$$ and we look for $x$ such that $y=123456$. Plotting the function, we can see that the solution is close to $6$. So let us use Newton method with $x_0=6$. The successive iterates would then be $$x_1=6.173815857$$  $$x_2=6.153702325$$ $$x_3=6.153371723$$ $$x_4=6.153371635$$ which is the solution for ten significant figures.
But, when the function is stiff, a more efficient way is to solve  $$\log(y) = \log(A + B * R^x + C * S^x)$$ Doing the same as above, using the same method and starting value, the successive iterates would then be $$x_1=6.153427858$$  $$x_2=6.153371635$$  which is the solution for ten significant figures.
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