Lesson 35: y= ln (x)

1. Lesson 35: y=ln(x)

2. Warm Up Preview:

3. Graph the following on calc…. • Use the window x[-1,2] and y [0,5] • Notices the relative position of each graph • Less than zero, at zero, greater than zero

4. Properties of y=exand its inverse • Always increasing • One to one (inverse exists) • Inverse of y=ex defined as y=ln(x) • . • Key points of y=ex are (0,1), (1, e), (2, e2) • What would the key points of y=ln(x) be? • Use the domain and range of y=ex to find the domain and range for its inverse (y=ln(x))

5. Graphing y=ln(x) • Graph of ln(x) is constantly increasing and concave down. • Let’s compare the graphs for y=ex and y=ln(x) • Use the 3 key points to graph each • Vertical Asymptote at x=0

6. Graph comparison

7. Example Graph: • Where is the vertical asymptote?

8. Answer

9. Example 2: Graph • First of all, what unique thing will happen with this graph? • There will be a y-axis reflection

10. Y-axis reflection • Comparison of y=ex and y=e-x • Comparison of y=ln(x) andy=ln(-x)

11. Example 2 Continued