Natural Logarithm as you’ve never seen it

Consider the function y = e raised to the power of x. This function has a very special property. To understand this property, consider the line tangent to this curve. Now consider the slope of this line. At every point along this curve, the value of the slope is exactly equal to the value of y at that point. Another way to say that the slope is equal to y is to say that dy / dx is equal to y. Instead of thinking of y as a function of x, let's think of x y. Another way of saying that y equals e raised to the power of x is to say that x is equal to the natural logarithm of y. The slope of the purple line is equal to 1 / y. — Let's show the function x = 1 / y as a green curve with a purple sphere traveling along it. The height of the purple sphere is always exactly equal to the slope of the purple line. As the height of the purple sphere approaches zero, the slope of the purple line approaches zero. As the height of the purple sphere approaches positive infinity, the slope of the purple line approaches positive infinity. But what about the portion of the function x = 1 / y where x and y are negative? All our previous relationships are still valid if we change the natural logarithm of y to the natural logarithm of the absolute value of y. — If the height of the purple sphere is a large negative number, the slope of the purple line is a large negative number. Heat.

— If the height of the purple sphere approaches zero, the slope of the purple line approaches zero. If the height of the purple sphere is a large positive number, the slope of the purple line is a large positive number. If the height of the purple sphere is a large negative number, the slope of the purple line is a large negative number. The height of the purple sphere is again always exactly equal to the slope of the purple line. In this video, we have only considered real numbers for x and y. An entirely new world opens up once we consider imaginary numbers and functions of complex variables.