How can we come up with the definition of natural logarithm?

I learned calculus for 2 years, but still don't understand the definition of $\ln(x)$

$$\ln(x) = \int_1^x \frac{\mathrm d t}{t}$$

I can't make sense of this definition. How can people find it? Do you have any intuition?

We want a function that changes multiplication into addition. That is, we want $$f(xy) = f(x) + f(y).\tag 1 $$

Substituting $y=1,$ we get $f(x) = f(x) + f(1),$ so we know that $f(1) = 0.$

Now, let's suppose that $f$ is differentiable. After all, we want to find as nice a function as possible. Let's hold $y$ constant for the moment, and differentiating $(1)$ gives $$yf'(xy) = f'(x) \implies \frac{f'(xy)}{f'(x)}=\frac{1}{y}$$

Now it's not hard to guess that $f'(x) = 1/x$ fills the bill, and together with $f(1)=0,$ the fundamental theorem of calculus gives us the definition.

Many calculus texts start with the exponential, and the fact that it is equal to its own derivative, and that it has an inverse function if the codomain is taken to be $(0,\infty)$. By the Inverse Function Theorem, if we write $f(x)=e^x$ and $b=e^a$, $$\tag1 (f^{-1})'(b)=\frac1{f'(a)}=\frac1{e^a}=\frac1b. $$ The inverse function of the exponential is usually named $\ln x$, and by $(1)$ we know that $(\ln x)'=1/x$. We also know that $\ln 1=0$, since $e^0=1$. Then $$\tag2\ln x=\int_1^x(\ln t)'\,dt=\int_1^x\frac1t\,dt. $$ The above shows that the natural logarithm should satisfy $(2)$.

Now, it is not easy to come up with the exponential in a constructive way, in particular at an elementary level. So it is easier to start with $(2)$, and construct the exponential as the inverse of $\ln x$.

At a more advanced level, one can start by defining $e^x$ via the Taylor series and then deducing $(2)$ as above. But that wouldn't cut it in a first calculus course.

The motivation to consider such a function can come from the following observation. We know that $$ \int x^n\,dx=\frac{1}{n+1}x^{n+1}\quad (n\neq-1). $$ Hence we may be interested in an antiderivative of $1/x$. Thus we consider the function (which we name) $$ \log(x)=\int_{1}^x\frac{1}{t}\,dt $$ which has the property that $(\log x)'=\frac{1}{x}$ by the fudamental theorem of calculus. From here we can derive its properties (such as $\log(xy)=\log(x)+\log(y), (x,y>0))$ and realize that the function is indeed the logarithm.