What is an intuitive way to see why harmonic series sums to logn?

the harmonic series is just a discrete approximation of that integral

Answer by Mark Eichenlaub:

It doesn't, but it grows asymptotically as $\log n$ because
$\log n \equiv \int_1^n \frac{1}{x} \mathrm{d}x$
and the harmonic series is just a discrete approximation of that integral.

What is an intuitive way to see why harmonic series sums to logn?