From A source book in Mathematics: Gauss (1971, at the age of fourteen) was the first one to suggest, in a purely empiraical way, the asymptotic formula $ \displaystyle \frac{x}{\log{x}}$ for $\phi(x)$.$^1$ Later on (1792-1793,1849) he suggested another formula $ \displaystyle \int_2^x \frac{dx}{\log{x}}$ of which $ \displaystyle \frac{x}{\log{x}}$ is the leading term$^2$. Legendre, being of course, unaware of Gauss' results, suggested another empirical formula $ \displaystyle \frac{x}{A \log x+B}$ $^3$ and specified the constants $A$ and $B$ as $A=1$, $B=-1.08366$$^4$ in the second edition of the Essai. Legendre's formula, which Abel quoted as "the most marvelous in mathemtics",$^5$ is correct up to the leading term only. This fact was recognized by Dirichlet$^6$. In this note Dirichlet suggested another formula $\displaystyle \sum^x \frac{1}{\log n}$.
$1$: (Werke, Vol.X$_1$, p.11, 1917).
$2$: (Gauss's letter to Encke, 1849, Werke, Vol II, pp.444-447, 1876)
$3$: (Essai sur ka théorie des nombres, 1st. ed., pp 18-19, 1789)
$4$: [Not from the book, but Wikipedia] He guessed $B$ to be about $1.08366$, but regardless of its exact value, the existence of $B$ implies the prime number theorem.
$5$: (letter to Holmboe, Abel Memorial, 1902, Correspondence, p.5)
$6$: ("Sur L'uasge des séries infinies dans la théorie des nombres." Crelle's Journal, Vol 18., p.272, 1838, in his remrk written on the copy presented to Gauss. Cf.Dirichlet, Werke, Vol. 1 , p372, 1889)
From Wikipedia: Initially, it might seem that since our numbering system is base 10, this base would be more "natural" than base $e$. But mathematically, the number $10$ is not particularly significant. Its use culturally—as the basis for many societies’ numbering systems—likely arises from humans’ typical number of fingers. Other cultures have based their counting systems on such choices as 5, 8, 12, 20, and 60.
$\log_e$ is a "natural" $\log$ because it automatically springs from, and appears so often in, mathematics.
Further senses of this naturalness make no use of calculus. As an example, there are a number of simple series involving the natural logarithm. Pietro Mengoli and Nicholas Mercator called it logarithmus naturalis a few decades before Newton and Leibniz developed calculus.