The Significance of Natural Logarithm in Modern Mathematics

1. Section 5.1 The Natural Logarithmic Function: “The miraculous powers of modern calculation are due to three inventions: The Arabic Notation, Decimal Fractions, and Logarithms.” – Florian Cajori, A History of Mathematics (1893) NPR

2. John Napier (1550-1617) • Invented Logarithms • Coined the term logarithm – “ratio number” • Spent 20 years developing logarithms • Published his invention in Mirifici Logarithmorum canonis descriptio (A description of the Marvelous Rule of Logarithms) NPR

3. Logarithms were quickly adopted by scientists all across Europe and China. • Astronomer Johannes Kepler used logarithms with great success in his elaborate calculations of the planetary orbits. • Henry Briggs, a professor of Geometry, later published table of logarithms to base 10 of all integers from 1 to 20,000 and from 90k to 100k in Arithmetica logarithmica. NPR

4. Properties: • Domain: ________ Range: ________ • Continuous, increasing, and one-to-one. • Concave ___________ NPR

5. Properties: • Domain: ___(0,∞)_ Range: ___(- ∞ , ∞ )_ • Continuous, increasing, and one-to-one. • Concave ___downward____ NPR

6. Logarithmic Properties If a and b are positive and n is rational, then the following properties are true: • ln(1) = • ln(ab)= • ln(a^n)= • ln(a/b)= NPR

7. Logarithmic Properties If a and b are positive and n is rational, then the following properties are true: • ln(1) = 0 • ln(ab)=lna + lnb • ln(a^n)=nlna • ln(a/b)=lna-lnb NPR

8. Expanding Log Expressions • ln(5/3)= • ln(4x/7)= NPR

9. The number e • The base for the natural logarithm • ln e = 1 • e is irrational • e ≈ 2.71828182846 • “The interest on a bank account, the arrangement of seeds in a sunflower, and the shape of the Gateway Arch in St. Louis are all intimately connected with the mysterious number e” –Eli Maor, The Story of a Number NPR

10. Evaluating Natural Log ExpressionsCalculator Active • ln 2= • ln 32= • ln 0.2= No-Calculator • ln e= • ln 1/e^3= • ln (e^n)= NPR

11. Using Properties: NPR

12. References • Larson, Hostetler, Edwards. Caclulus of a Single Variable.7th Edition.New York: Houghton Mifflin Company, 2002. • Maor, Eli. e: The Story of A Number.New Jersey: Princeton University Press, 1994. NPR