calculus - 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?

Answer

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.