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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.

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  • $\begingroup$ Well, logarithms in general were an useful tool for computing. I guess they came in first. Then probably someone saw that we can use $\int_1^x\frac{1}{t}dt$ as an alternative definition of $\ln(x)$. $\endgroup$ – Jakobian Oct 5 '18 at 8:14
  • $\begingroup$ I'm sure that $y = \log_b(x)$ was introduced as being the solution to $b^y = x$. With this definition, it's straightforward to discover the properties of logarithms such as $\log_b(xy) = \log_b(x) + \log_b(y)$. I guess the number $e$ was originally introduced as the special value of $b$ that makes the derivative of $b^x$ equal to $b^x$. It was then discovered (I imagine) that the derivative of $\log(x)$ is $1/x$, and so $\log(x) = \int_z^x 1/t \, dt$. $\endgroup$ – littleO Oct 5 '18 at 8:16
  • $\begingroup$ $e^x$ appears to be injective and has range $(0,\infty)$. Reason enough to define its inverse and "baptize" it with some name (let's go for "log"). Then based on the property of the exponential function that you mention it can easily be deduced that $\log(xy)=\log x+\log y$. Also it is discovered that $\log x=\int^x_1\frac1tdt$. That is not the real history of course, but everything in it sounds reasonable (except maybe that peculiar name "log") . $\endgroup$ – drhab Oct 5 '18 at 8:16
  • $\begingroup$ If you want historical perspective, then perhaps you should ask this question on the History of Science and Mathematics Stack Exchange. BTW: one can show that $\log(xy)=\log x+\log y$ from clever manipulation of the integral definition. $\endgroup$ – Blue Oct 5 '18 at 8:17
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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

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