The slope of a tangent to a logarithm curve

Drag the slider to move the point at which the tangent is being computed.

Move points A₁ and H to adjust the shape of the logarithm curve.
To see the same results as a table, drag the divider to see the spreadsheet window at right.

The exact slope of the tangent to a curve that is defined algebraically as the graph of a function can be computed with calculus by taking the derivative. This diagram shows how to get the approximate slope, to as much precision as you like, without calculus. The points on the curve were plotted using Descartes's logarithm machine.

Calculus uses the slope calculated using the center point and a point to the right or left, then takes a limit. But before you take the limit, the slope using the left and right points is much more accurate.
See Descartes's Logarithm Machine for a full explanation.

Related demonstration files:

Similar right triangles

Geometric means

Descartes's mean proportionals/logarithm machine

Descartes's machine used to solve a "two mean proportionals" problem

Descartes's machine used to plot points on the graph of a logarithm function

Descartes's construction used to find the slope of a tangent to a logarithmic graph

Susan Addington, Created with GeoGebra

Quadrivium

Applets