Let's look at the area under 1/x from 1 to infinity. Place a rectangle of width 1 and height 1/2 under the curve, as x runs from 1 to 2. Place a second rectangle of width 2 and height 1/4 under the curve, as x runs from 2 to 4. Place a third rectangle of width 4 and height 1/8 under the curve, as x runs from 4 to 8. Continue doing this forever. Each rectangle has area 1/2, hence the area under the curve is infinite. The log(x) approaches ∞ as x approaches ∞. By symmetry, the area under 1/x, from 0 to 1, is also infinite. Thus log(x) approaches -∞ as x approaches 0.
By construction, the derivative of log(x) is 1/x. As the log curve crosses the x axis at x = 1, its slope is 1. The slope flattens out as x approaches infinity, yet the function is always increasing. As you move back to x = 0, log(x) plunges down the y axis, and becomes almost vertical.