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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). John Napier (1550-1617). Invented Logarithms - PowerPoint PPT Presentation
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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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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)
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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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Properties:
1) Domain: ________ Range: ________
2) Continuous, increasing, and one-to-one.
3) Concave ___________
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Properties:
1) Domain: ___(0,∞)_ Range: ___(- ∞ , ∞ )_
2) Continuous, increasing, and one-to-one.
3) Concave ___downward____
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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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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
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Expanding Log Expressions
ln(5/3)=
ln(4x/7)=
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The number e
The base for the natural logarithmln e = 1e 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
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Evaluating Natural Log ExpressionsCalculator Active
ln 2=
ln 32=
ln 0.2=
No-Calculator
ln e=
ln 1/e^3=
ln (e^n)=
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Using Properties:
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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.