Natural Logarithm

Some book authors define the natural logarithm function, ln (x) as


If we take a look at the graphs below, the red curve represents the graph of , and the dark green curve represents the graph of . The purple region falls below the red curve is bounded between x = h and 1, while the blue region is bounded between x = 1 and k.

Now scroll the value of k, do you notice that point K moves according to the k value? The y-coordinate of point K is the area of the blue region. Same concept applies to the value of h and point H. Here comes the question, do you know how to get negative y-coordinate for point H?



Betty, Created with GeoGebra

Spiral

A spiral is formed with a simple equation . The graph below set 'a' as positive, therefore there are two equations for you to tick. Choose whichever equation you prefer, you may choose both as well, then drag the value of 'a' according to your preference. Then, drag the value of to view the forming of spiral.

Betty, Created with GeoGebra