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.
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