This applet uses the base a exponential curve, the line y=x, and the notion of a "cobweb" diagram to construct a point on the base a logarithm curve. The base a can be adjusted using the slider. By moving the point labeled "A" on the x-axis you can trace out a set of points on the log curve.
Check the box to display the base a log curve, and see that it hits all of your traced points. Use the rightmost menu button to wiggle the picture and erase traces.