Suppose:
$$a = bw^f + cw^g $$
where $a,b$ and $c$ are known, and $f$ and $g$ are known fractional exponents
Ex. $50000 = 200w^{0.72} + 4000w^{0.19}$
How can one solve for the value of w?
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Except if very particular cases, equations such as $$f(w)=b w^f+c w^g-a$$ do not have analytical solutions and numerical methods should be used. One of the simplest root-finding methods is Newton which, starting from a reasonable guess $w_0$, will update it according to $$w_{n+1}=x_n-\frac{f(w_n)}{f'(w_n)}$$ For your example, a quick look at the graph of the function shows that (let me be very lazy) there is a root betwenn $1000$ and $2000$. So, let us start Newton with $w_0=1000$; the method then generates the following iterates : $1263.55$, $1275.06$, $1275.08$ which is the solution for six significant figures.
Claude Leibovici
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