Title: What is ln(x)?

Author: Amanda Robertson

Topic: The Natural Logarithm

Connection to Core Curriculum: CCSS.Math.Content.HSF-LE.A.4: For exponential models, express as a logarithm the solution to ab^ct=d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
Overview: Students will use an applet to learn the interpretation of ln(x) as giving the area under the curve 1/x and above the interval [1,x].
Objectives: Students will learn that the natural log gives the value of the integral ∫₁^x1/t dt.
Students will learn why ln(1)=0 and ln(e)=1.

Materials Needed: Applet: What is ln(x)?

Activity Plan:
For this activity students will use the applet to learn what ln(x) means. The applet shows the definition of ln(x) in terms of an integral for 1/t. Students can see that the natural logarithm gives the value for the area under the curve 1/x over a specified interval. Students can use the sliders to change the value of x to see why ln(1)=0 and to find that ln(e)=1. Some questions to ask (these are also on the applet) are: for what value of x is the area under the curve 1/x equal to 1? Why does ln(x) give the negative of the area if 0

Included Documents: Worksheet
Using the applet, find the following values:
1. ln(3)
2. ln(1.34)
3. ln(.67)

4. What do you notice about the answer to question 3. that is different from those for questions 1. and 2.? Use the applet to find out why it is different.



5. What does ln(1) equal? Why do you think that is? (Write 2 sentences about what you think and then use the applet to see why ln(1)=0)



6. For what value of x is the area under the curve equal to 1? What do you know about this value?