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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.