# NPR1 Section 5.1 The Natural Logarithmic Function: “The miraculous powers of modern calculation are due to three inventions: The Arabic Notation, Decimal.

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NPR1 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)

NPR2 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)

NPR3 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.

NPR4 Properties: 1)Domain: ________ Range: ________ 2)Continuous, increasing, and one-to-one. 3)Concave ___________

NPR5 Properties: 1)Domain: ___(0,∞)_ Range: ___(- ∞, ∞ )_ 2)Continuous, increasing, and one-to-one. 3)Concave ___downward____

NPR6 Logarithmic Properties If a and b are positive and n is rational, then the following properties are true: 1)ln(1) = 2)ln(ab)= 3)ln(a^n)= 4)ln(a/b)=

NPR7 Logarithmic Properties If a and b are positive and n is rational, then the following properties are true: 1)ln(1) = 0 2)ln(ab)=lna + lnb 3)ln(a^n)=nlna 4)ln(a/b)=lna-lnb

NPR8 Expanding Log Expressions ln(5/3)= ln(4x/7)=

NPR9 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

NPR10 Evaluating Natural Log Expressions Calculator Active ln 2= ln 32= ln 0.2= No-Calculator ln e= ln 1/e^3= ln (e^n)=

NPR11 Using Properties:

NPR12 References Larson, Hostetler, Edwards. Caclulus of a Single Variable.7 th Edition.New York: Houghton Mifflin Company, 2002. Maor, Eli. e: The Story of A Number.New Jersey: Princeton University Press, 1994.

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