<p>I have a function f(x) in which there is log function </p>

<p><span class="math-container">$f(x)=log(1+ax)-bxg(x)$</span></p>

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

<p>I tried <span class="math-container">$g(x)=\frac{1}{1+cx}$</span> but it becomes too complex (quadratic).</p>

<p>Similarly I tried <span class="math-container">$g(x)=exp(cx)$</span> but that too turns out to be a lambert function. </p>

<p>Any guesses for a simple discount function g(x). </p>

CC BY-SA 4.0
2019-10-09T08:34:11.940Z
What do you mean by "value of $x$ is not too complex"? Isn't $x$ just $x$?
Art
CC BY-SA 4.0
2019-10-09T11:33:54.940Z
I mean after derivating  f(x) , i need to get the value of x
user7341333
CC BY-SA 4.0
2019-10-09T14:32:39.830Z
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
CC BY-SA 4.0
2019-10-10T05:10:51.173Z
yes a,b,x take on positive values
user7341333
CC BY-SA 4.0
2019-10-10T07:15:25.533Z
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
<discount>
discount function with logarithm
user7341333
# discount function with logarithm

(Asked by: username_0 on 2019-10-09)

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).

**Tags:** discount

> **Comments:**
> **username_1**: What do you mean by "value of $x$ is not too complex"? Isn't $x$ just $x$?  
> **username_0**: I mean after derivating  f(x) , i need to get the value of x  
> **username_2**: Use the fact that the sum of concave functions is concave. What do you know about $a,b,x$, do they take positive values?  
> **username_0**: yes a,b,x take on positive values  
> **username_2**: 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?  

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## 0 answers

