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Is it possible to solve this functional equation?

$$\frac{ln(a[t = t^*])}{ln(a[t])}=f[t]$$

where $a[t]$ is the unknown function of the independent variable $t$; $f[t]$ is known, and $t^*$ is a specific value of $t$, also known.

If yes, how would you do? Do you think that other information is needed? If there is not an analytically closed form, are there any numerical methods to estimate $a[t]$?

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Rewriting, $\ln(a(t)) = \dfrac{\ln(a(t^*))}{f(t)}=\ln\big(a(t^*)^{1/f(t)}\big)$ and $$a(t) = a(t^*)^{1/f(t)}.$$ In particular, $a(t^*) = (a(t^*))^{1/f(t^*)}$. This holds only when $f(t^*)=1$. Without that condition, there is no solution.

Note that you could have seen this from the outset. Set $t=t^*$ in the given equation and you immediately get $f(t^*)=1$. But, assuming that hypothesis, I've given you the formula for $a(t)$ in general.

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  • $\begingroup$ Should be $a(t)=$. Otherwise, just what I would do. $\endgroup$ – marty cohen Apr 7 at 18:34
  • $\begingroup$ @Ted Thank you for the rapid and clear answer! Hence there is no way of knowing the value of $a$ at time $t=t^*$ just from this equation, right? $\endgroup$ – Shootforthemoon Apr 7 at 19:29
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    $\begingroup$ Right. You will get a different solution for each (positive) value at $t^*$. $\endgroup$ – Ted Shifrin Apr 7 at 19:31

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