The number "e" was discovered when calculating the compound interest in the 17th century, which was more than half a century before calculus was invented. And the function y=e^x was used to calculate the compound interest. Since it is an "exponent" function whose base is "e", it is intuitive to create "logarithm" function for it, which does the reverse calculation. We call it "ln(x)". But so far the "ln(x)" was just a symbol. After the invention of calculus, people discovered that the integral of 1/t between 1 and x is the reverse function of e^x. This astonishing fact allows us to give the symbol "ln(x)" a formula that is calculatable by itself. And this is why sometimes it is said to be the definition of "ln(x)". But actually, the real timeline is quite reversed.
