I found that the logarithmic derivative can also be given by first principles like this:

$$\lim_{h\rightarrow 1}\log_{h}\frac{f(xh)}{f(x)}$$

From this, I got this approximation involving the first logarithmic derivative of the function:

$$f(x)\approx f(a)\left(\frac{x}{a}\right)^{f'(a)}$$ where $f'(a)$ is the first logarithmic derivative of $f$ at $x=a$.

Similarly, this should be the approximation involving the second order logarithmic derivative:

$$f(x)\approx f(a)e^{\frac{f'(a)}{a^{f''(a)}(f''(a)+1)}(x^{f''(a)+1}-a^{f''(a)+1})}$$ Here, $f'(x)$ and $f''(x)$ are the first order and second order logarithmic derivatives respectively. These approximations are supposed to work when $\frac{x}{a}\approx 1$.

First, Have I got these two results correct? And second, Is it possible to give an infinite series involving logarithmic derivatives? I mean, an infinite series is definitely possible, but is it possible to give a general formula of that infinite series? I see no pattern here. The second approximation looks ugly.


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