calculus - How can I prove the integral $int_{1}^{x} frac{1}{t} , dt$ is $ln x$ with this approach?

I have been trying to find a proof for the integral of $\int_1^x \dfrac{1}{t} \,dt$ being equal to $\ln \left|x \right|$ from an approach similar to that of the squeeze theorem.

Is it possible to calculate the area under the curve $f(x) = \dfrac{1}{x}$ as in the picture shown below? You may notice that both sums of the areas should converge to $\ln(x)$ as the base of the rectangles gets smaller and smaller. We approach from above and from below to get a limiting argument of the form:

Area from below the curve as in the second graph $\leq \int_a^b \dfrac{1}{x} \,dx \leq$ Area from above the curve as in the first graph

The limiting argument would be to keep calculating and adding the areas of the rectangles which bases get smaller and thus showing that this amount is $\ln(x)$.

How could I proceed in this way?