How to solve $f(x)$ in the equation $f(x)/F(x)=\ln(2)$

Question

I need to give one example of what $f(x)$ can be in the equation $\frac{f(x)}{F(x)}=\ln(2)$. $F(x)$ is the primitive function to $f(x)$. I need help to understand how to do that.

Answer 1

$\dfrac {f(x)} {F(x)} = \ln 2$ means that $F(x) = \dfrac {f(x)} {\ln 2}$, which you can rewrite as $F(x) = \dfrac {\frac {{\rm d} F} {{\rm d} x}} {\ln 2}$, the latter being a differential equation that you can solve: $F(x) = {\rm e} ^{x \ln 2}$.

Answer 2

Letting $g(x)=F(x)$ we see this gives $\ln(2) g(x)=g'(x)$. You can separate and integrate, or you can remember that this is the equation that defines $f(x)=k2^x$, for $k\neq 0$ due to the division in the original problem.