History of logarithmic function

Question

The exponential function is a very important function and it arises naturally.

For instance, consider the limit $\displaystyle \lim_{n \to \infty} (1+\dfrac{1}{n})^n$.

The limit is evaluated to be the real number $2.718281\dots$ which is denoted by $e$. Another limit $\displaystyle \lim_{n \to \infty} (1+\dfrac{x}{n})^n$ is evaluated to be $e^x$.

Okay, it's easy to check by the binomial expansion of $\displaystyle \lim_{n \to \infty} (1+\dfrac{x}{n})^n$ that $f(x)=e^x$ satisfies the property $f(x)f(y)=f(x+y)$.

But how did we come to know that the inverse function of $e^x$ is $\log x$? Introducing $\log x$ as $\displaystyle \int_1^x \dfrac{1}{t} \ dt$ is unintuitive and it doesn't tell any property the function has.

For instance how did we come to know $f(x)=\log x$ satisfies the property $f(xy)=f(x)+f(y)$?

And what was the motivation for introducing the logarithmic function? It doesn't make calculation any easier.

Answer 1

In 1649, Alphonse Antonio de Sarasa, a former student of Grégoire de Saint-Vincent, related logarithms to the quadrature of the hyperbola, by pointing out that the area $A(t)$ under the hyperbola from $x = 1$ to $x = t $ satisfies ${\displaystyle A(tu)=A(t)+A(u).}$

Historian Tom Whiteside described the transition to the analytic function as follows

By the end of the 17th century we can say that much more than being a calculating device suitably well-tabulated, the logarithm function, very much on the model of the hyperbola-area, had been accepted into mathematics. When, in the 18 century, this geometric basis was discarded in favour of a fully analytical one, no extension or reformulation was necessary – the concept of "hyperbola-area" was transformed painlessly into "natural logarithm".

Source: https://en.wikipedia.org/wiki/History_of_logarithms

Mathematical Association of America: Articles on History of Logarithm