disclamer: there is no rule called exponential rule and logarithmic rule, they are just called "derivative of \(\log\)" or anything else, I just call them that cuz it sounds cool :3 + u can prove any differentiation identity using the standard formula: \(\lim_{h \to 0}\frac{f(x+h) - f(x)}{h}\)
I think u are familar with an exponential function: $$f(x) = c^x$$ the derivative of this function is: $$\frac{d}{dx} f(x) = c^x \cdot \ln(c)$$ this also indicates that the only thing that is equal to its own derivative is \(e^x\) because \(\ln(e) = 1\) so: $$\frac{d}{dx} e^x = e^x$$
$$\frac{d}{dx} \log_c(x) = \frac{1}{x \cdot \ln(c)}$$ so by the same logic $$\frac{d}{dx} \ln(x) = \frac{1}{x}$$