Explore the gradient function of natural log ([math]\ln(x)[/math]) by considering the connection between the exponential function and the natural log function. The graphs displayed are [math]y=e^x[/math], [math]y=x[/math], and [math]y=ln(x)[/math]. In addition, point P [math](x_1,y_1)[/math] on [math]y=e^x[/math] and point Q [math](y_1,x_1)[/math] on [math]y=ln(x)[/math] are shown. Tangents are shown at these points and the gradients of the tangents are displayed.
1. What transformation maps the graph [math]y=e^x[/math] to the graph [math]y=\ln(x)[/math]? Adjust the point P using the slider for p. 2. At any point on the graph, if the gradient of y=e^x is m, what is the gradient of y=ln(x)? ([i]Hint[/i]: think about the dimensions of the triangles.) 3. Work out the gradient function for [math]y=ln(x)[/math]. (At the point [math](x,y)[/math], the gradient of [math]y=e^x[/math] is [math]e^x = y[/math]. So at the point [math](y,x)[/math], the gradient of [math]y=\ln(x)[/math] is ... So at the point [math](x,y)[/math], the gradient of [math]y=\ln(x)[/math] is ...