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I have a function f(x) in which there is log function

$f(x)=log(1+ax)-bxg(x)$

I need a discount function g(x) with which the function f(x) is concave and the value of x is not too complex.

I tried $g(x)=\frac{1}{1+cx}$ but it becomes too complex (quadratic).

Similarly I tried $g(x)=exp(cx)$ but that too turns out to be a lambert function.

Any guesses for a simple discount function g(x).

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    $\begingroup$ What do you mean by "value of $x$ is not too complex"? Isn't $x$ just $x$? $\endgroup$ – Art Oct 9 at 8:34
  • $\begingroup$ I mean after derivating f(x) , i need to get the value of x $\endgroup$ – user7341333 Oct 9 at 11:33
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    $\begingroup$ Use the fact that the sum of concave functions is concave. What do you know about $a,b,x$, do they take positive values? $\endgroup$ – Bertrand Oct 9 at 14:32
  • $\begingroup$ yes a,b,x take on positive values $\endgroup$ – user7341333 Oct 10 at 5:10
  • $\begingroup$ In this case, you can consider any increasing and convex function $g$ like $g(x)=x$ for instance or $g(x)=\exp(x)$ (some decreasing and convex functions should also work fine, locally). Derivation is feasible with most functions, and if you want to solve something like $f'(x)=0$, this is analytically and numerically possible for a broad range of functions. You should clarify what you mean by "I need to get the value of $x$". How is this value characterized? $\endgroup$ – Bertrand Oct 10 at 7:15

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