discount function with logarithm

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).

• What do you mean by "value of $x$ is not too complex"? Isn't $x$ just $x$? – Art Oct 9 at 8:34
• I mean after derivating f(x) , i need to get the value of x – user7341333 Oct 9 at 11:33
• Use the fact that the sum of concave functions is concave. What do you know about $a,b,x$, do they take positive values? – Bertrand Oct 9 at 14:32
• yes a,b,x take on positive values – user7341333 Oct 10 at 5:10
• 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? – Bertrand Oct 10 at 7:15