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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)

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NPR2 John Napier ( ) 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)

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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.

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

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

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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)=

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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

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

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NPR9 The number e The base for the natural logarithm ln e = 1 e is irrational e ≈ “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

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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)=

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NPR11 Using Properties:

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

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