Using log to take derivative of a function

Question

Is it safe to say that if $\frac{d}{dx}ln(f)= g $ for some functions f and g, then $\frac{d}{dx}f = e^{g}$? Why or why not?

(novice high schooler here)

Answer 1

The answer is "no" as pointed out in the comments. You can however do the following:

$\frac{d}{dx} \ln(f(x)) = \frac{1}{f(x)}\cdot f^{\prime}(x)$ by the chain rule. So, if $\frac{d}{dx}\ln(f(x))=g(x)$ then $f^{\prime}(x)=f(x)g(x)$. This is usually called "logarithmic differentiation" and tends to show up in text books near applications of derivatives or implicit differentiation.