I know that :
$$\ln: (0,+\infty )\to \mathbb{R}$$ $$\ln(x):=\int_1^x \dfrac{dt}{t}$$
And :
$$\ln^{-1}: \mathbb{R} \to \mathbb{R}^+$$ $$\ln^{-1}(x):=\exp(x)$$
after that. We define :
$$f: \mathbb{R} \to \mathbb{R}^+$$ $$f(x):=\big(\exp(\ln a)\big)^x \ \ \ a >0 ,a \neq 1$$
And :
$$f^{-1}: \mathbb{R}^+ \to \mathbb{R}$$ $$f^{-1}(x):=\log_a (x) \ \ \ a >0 ,a \neq 1$$
now my question : function $\dfrac{1}{t}$ How did they get?
The discovery of (decimal) logarithms by Napier (tables published in 1614) predates by a century or so the discovery of the fact that the area between $1$ and $a$ under the hyperbola with equation $y=\dfrac{1}{x}$ has a value which is proportional to a multiple of the (decimal) logarithm of $a$. It was not until the second half of the 18th century, with the fixation of notations by Euler, who in particular introduced the natural logarithm (also called Euler logarithm, now denoted $\ln$) that it became current to write
$$\int_a^b \dfrac{dx}{x}=\ln(b)-\ln(a)$$
(the integral notation was introduced only 50 years earlier, by Leibnitz).
It was the same Euler who fixed the notation $e^x$ for the exponential function, and explained in a plain way that exponential and logarithm are inverse functions one of the other.